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Accessions  Np..jL*$~~fy/'S~       Shelf  No. 


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TREATISE 


ON 


ALGEBRA, 


FOR    THE   USE   OF 


SCHOOL^"  AND^(?ODLEGES. 


BY 

S.   CHASE, 

PROFESSOR    OF    MATHEMATICS   IN    DARTMOUTH    COLLEGE. 

row 

NEW-YORK: 
D.  APPLETON  &  CO.,  200  BROADWAY. 

PHILADELPHIA  : 
GEO.    S.    APPLETON,    164  CHESNUT-ST. 

MDCCCXHX, 


Entered  according  to  Act  of  Congress,  in  the  year  1849,  by 

S.  Chase, 

In  the  Clerk's  office  of  the  District  Court  of  the  District  of  New 

Hampshire. 

I  tiff  ST 


DARTMOUTH   PRESS, 
Hanover,  If.  H. 


PREFACE 


The  following  treatise  is  intended  to  exliibit  such  a  view  ot 
the  principles  of  Algebra,  as  shall  best  prepare  the  student  for 
the  further  pursuit  of  mathematical  studies. 

The  principles  presented  I  have  endeavored  to  enunciate  as 
clearly  and  briefly  as  possible,  to  demonstrate  rigorously,  and  to 
illustrate  by  strictly  pertinent  examples. 

Part  of  the  examples  are  of  the  most  elementary  form,  part 
are  purely  numerical,  and  a  large  part  of  the  rest  are  expres- 
sions employed  in  the  reasonings  and  investigations  of  Trigo- 
nometry, Analytical  Geometry,  Mechanics  and  other  branches  of 
mathematical  study.  Thus,  the  application  of  the  principle  is 
exhibited,  relieved  of  all  extraneous  difficulty,  and  connected 
with  the  familiar  ideas  of  Arithmetic ;  and,  moreover,  the  forms 
and  operations  employed  in  demonstrating  truths  of  Geometry, 
and  of  other  related  sciences,  are  rendered  familiar,  and  made 
ready  for  use  when  they  shall  be  needed. 

This  last  consideration  is  of  great  importance.  Much  of  the 
difficulty  which  students  find  in  later  parts  of  the  course  results 
from  want  of  familiarity  with  the  algebraic  expre'ssions  employed, 
and  with  the  elementary  operations  performed  upon  them.  At 
the  same  time,  such  expressions  and  operations  are  frequently 
among  the  most  convenient  illustrations  of  algebraic  principles. 

The  discussion  of  the  theory  of  exponents  and  powers 
(§§  11-24)  is,  so  far  as  I  know,  original.    The  use  and  interpre- 


IV  PREFACE. 

tation  of  the  fractional  and  negative  exponents  is  exhibited  as  a 
necessary  consequence  of  the  definition. 

The  demonstration  of  the  Binomial  Theorem  for  negative  and 
fractional  exponents  (§§  291-294),  and  the  development  of  the 
fundamental  logarithmic  formula  (§§  320-323)  are  substantially 
those  of  Lagrange. 

The  nature  of  the  Modulus  (§§327-332),  and  some  of  the 
properties  of  logarithmic  differences  (§§333-336)  are  discussed 
more  fully  than  I  have  seen  them  in  any  elementary  treatise. 
Familiarity  -with  these  principles  is  of  great  advantage  to  the 
student,  and  their  discussion  is,  by  no  means,  difficult. 

A  table  of  the  principal  formulae  of  the  book  is  placed  after 
the  table  of  contents,  for  convenience  of  reference  and  review. 
It  has  also  the  advantage  of  generalizing,  and  bringing  into  one 
view,  principles  exhibited,  -with  more  or  less  fulness,  in  different 
parts  of  the  book.  For  the  suggestion  of  this  table,  I  am  indebt- 
ed to  Mr.  Richards,  the  able  Principal  of  Kimball  Union  Acad- 
emy. 

I  am  also  very  greatly  indebted  to  my  associates,  Professors 
Crosby  and  Young,  for  valuable  suggestions  and  criticisms.  In 
correcting  the  proofs  of  the  last  half  of  the  work,  I  have  had  the 
assistance  of  Mr.  Edward  Webster,  a  recent  graduate  of  the  Col- 
lege, whose  tastes  and  attainments  qualify  him  to  do  excellent 

service  in  the  cause  of  science. 

S.  C. 
Dartmouth  College,  May  1,  1849. 


CONTENTS. 


INTRODUCTION. 

Symbols  of  quantity, — Signs, 13-i< 

Positive  and  negative  quantities,            ....  17-21 

Factors, — Coefficient, — Exponent,     .        .        .  22-25 

Exponent ;  fractional,  zero,  negative, — Reciprocal,  .     26-34 

Power, — Root, — Function, 34.-4* 

Degree, — Homogeneous, — Like,     ...  40,41 

Monomials, — Binomials, — Polynomials,       ...  42 

Reduction  of  Polynomials, 43-45 

Equations ;  identical,  absolute,  conditional,    .         .  .     45, 4G 

Degree, — Axiom, — Transposition,      ....  47-50 

Clearing  of  fractions, — Solution, — Problem,            .  .     51-58 


CHAPTER  I. 

ADDITION  AND  SUBTRACTION. 

i.  Addition, 

ii.  Subtraction, — Combination  of  signs, 


59-62 
62-67 


CHAPTER  II. 

MULTIPLICATION  AND  DIVISION. 

in.  Multiplication, — Monomials, — Signs, — Degree,     -  68-69 

iv.  Polynomials, — Detached  coefficients,        .        .        .  70-75 

Division, — Monomials, — Signs,            ....  75-77 

Polynomials, — Detached  coefficients,     ....  77-83 

Synthetic  division, — Infinite  series,     ....  83-86 

Theorems,  (a  ±b) 2,  (a  +  b) (a  —  b),               .         .  86-89 

Divisibility,  (o«  —  bn)  -±  [a  —  b),  &c,    .         .         .  90-92 

Greatest  common  divisor, 92-96 

Least  common  multiple,  —  Problems,  %,  #,  .         96-100 

*1 


VI  CONTENTS. 


CHAPTER  III. 

FRACTIONS-  .  .  .  101 

Reduction, — Addition  and  Subtraction,      .         .         .  102-105 

Multiplication  and  Division,  ....  106-108 

CHAPTER  IV. 

EQUATIONS  OF  THE  FIRST  DEGREE,  WITH  TWO  OR 

MORE  UNKNOWN  QUANTITIES,  .  .  .  109 

Two  unknown  quantities,        .         .  .  109-115 

Elimination, — By  Addition  and  Subtraction,      .         .  110-112 

By  Comparison, — By  Substitution,  .         -         .  112,113 

More  than  two  unknown  quantitieSi  .         .         .  115-119 

«      »     p,  120-121 

CHAPTER  V. 

INEQUALITIES.  .  .  122-124 

CHAPTER  VI. 
POWERS  AND  ROOTS. 

Monomials,— Powers,— Roots,         ....  124-127 

Radical  quantities,— Imaginary  quantities,  .         .  127-131 

Polynomials, — Powers, ••  131-133 

Square  root  of  a  polynomial, — Of  a  number,      .         .  133-139 

Cube  root  of  a  polynomial, — Of  a  number,      .         .  139-142 

»»**  root  of  a  polynomial, 142 

Square  root  of  a  ±  5*, 143-147 

Binomial  surds  rendered  rational,       ....  147-149. 


CHAPTER  VII. 

EQUATIONS  OF  THE  SECOND  DEGREE. 

Definition,— Complete  equation,— Incomplete,  149-151 

Solution  of  incomplete  equation,         ....  151-15  j 

Solution  of  complete  equation,        .         •  156-160 


CONTENTS.  Til 

General  Discussion, — Factors, — Roots, — Signs,           .  161-164 

*«» -j- Par" -|- Q  =  0,— Radicals,            .          .          .  167_lt>9 

Two  unknown  quantities, — Variables, — Curves,      .  169-174 

CHAPTER  VIII. 

RATIO  AND  PROPORTION.           .  .           17c 

Mean  proportional, — Equal  Products,     .         .         .  176-178 

Inversion, — Alternation, — Composition, — Division,     .  178-180 

Equimultiples, — Sums, — Powers, — Products,           .  180-182 

I u verse  Proportion, — Variation,         ....  182-18C 

CHAPTER  IX. 

EQUIDIFFERENT,  EQUIMULTIPLE,  AND  HARMONIC  SERIES. 

i.  EquidifFerent  series,-r-Difl'erence,        ...  186 

Last  term, — Sum, — Mean, — Interpolation,          .         .  IS 7-1 90 

II.  Equimultiple  series, — Multiplier,        ...  190 

Last  term, — Compound  Interest, — Mean, — Sum,        .  191-194 

m  <  1  and  n  =  co, — Annuities,                    .          .  195-199 

Interpolation, 199 

in.  Harmonic  Series, — Proportion,    ....  200 

CHAPTER  X. 

PERMUTATIONS  AND  COMBINATIONS.  203-207 

CHAPTER  XI. 

UNDETERMINED  COEFFICIENTS.           .  208-212 

Development  of  series,    .....  210 

Decomposition  effractions, 211 

CHAPTER  XII. 

BINOMIAL  THEOREM. 

i.  Positive  integral  exponent,          ....  213-218 

ii.  Negative  and  fractional  exponent,         .         .         .  218-226 

Derived  fuuctlon,  or  derivative,      .         .         .  220-223 


Ylli  CONTENTS. 

CHAPTER  XIII. 

DIFFERENCES. 

Orders, — n'h  term, — Interpolation, — Sum,      .        .  227-234 

CHAPTER  XIV. 

INFINITE  SERIES.          .           .  235-24*' 

CHAPTER  XV. 

LOGARITHMS. 

Exponents, — Characteristic, — Base,        .        .        .  241-244 

Development, — Formulas, 246-250 

Modulus, — Naperian  system, — Differences,     .  250-257 

Computation, •         .  257 

Exponential  Theorem  and  Equation,      .        .        .  259-264 

CHAPTER  XVI. 

THEORY  OF  EQUATIONS. 

Degree,  integral,  fractional, 265 

Divisibility, — Roots, — Number  of  roots,           .         .  266-271 
Coefficients, — Form  of  the  roots,  fractional,  imaginary,  271-275 

Signs  of  the  roots, — Variations  and  permanences,      .  276-280 
Transformation, — Change  of  roots, — Removal  of  terms,  280-289 

Limits  of  the  roots,          ......  290-293 

Limiting  equation, — Equal  roots,        ....  293-296 

Sturm's  theorem,             296-306 

Numerical  equations, — Integral  roots,        .         .         .  306-309 

Incommensurable  roots, — Horner's  method,    .         .  309-318 

Recurring,  or  reciprocal  equations,    ....  318-322 

Biroinial  equations, 322-325 

CHAPTER  XVII. 

CONTINUED  FRACTIONS.            .  .           32C 

Convergents, — Error, — Lowest  terms,            .        .  327-331 

Reducing  to  a  continued  fraction,           .        .        .  332-335 


I TT  I T  Y 

FORMULA. 


§  6.   a.)    -}-  a  >  0  ;  —  a  <  0.     That  is, 

A  positive  quantity  >  0  ;  a  negative  quantity  <  0. 
§  7.    a,  b.)    —  (—  a)  =z -\- a.         .:  —  [-—  (— a)] =r  —  a ; 
—  (—(—(— «)))  =  +  «;  &c.     §§63;  68.  a,  <?. 
§13.  'a°=l.     §17.   a-n=  _. 

n  JL         n  »  ™ 

§  23.    ya  =  a",     y a'»  =  (Ja)m  =  a\     §§  1 2,  25. 

§57.3.     i(«  +  5)+K^  —  *)'=«■ 

§  60.  4.     £(a  +  5)  —  i(>  —  b)  —  b. 
§§89,90.    (a±b)*  =  a2±2ab-{-  62. 
§91.    (a+.Z>)2  +  («  —  £)2  =  2(a2  +  &2). 

(a  +  5)2  —  (a  —  6)2  —  4a5.     . 
§92.    (a  +  b)(a  —  b)  =  a2  —  62. 


§96.   a.) =  a»-i-j_a»-9J  .  .  .  -f-a&*-2  +  b"~l. 


.  ,   an— a"  „  , 

a  —  a 


§97.    -""  ,  f""  =  a2"-1—  a2"-s6+  .  .  4-a62"-2— &2"-i. 
a-}-6 

a2»+l    I    £2«+l 
§98.     4-= =  a2"—  a9t-lJ-l_.    .  _aJ2«-l   I    JSn, 


X  FORMUL-ffi. 

§§  109,  139,  140.)    pr  =  co.     —  =  0.     -,  indeterminate. 

§151,  c.)    (an)m  =  amn.     (±a)2"  =  -f-(a2»). 

(±a)2"+i  =  ±(«2»+i). 

i  1  i 

§  152.     (+  a)*n=± (a2",).     (—  «) 2",  imaginary. 

i  i 

(±«)2;i+i  =  ±  («2h-i). 

1158.  (-o2f  =  a(-l)^ffly-l. 
§162.     (y-l)2=-l;  (y-l)3=_y_i 

(</-l)*=l;    (y-i)5  =  y-i. 

§185.     (.±Ji)^(^±(^; 

where  c  =  (a2  —  J)2. 

§  186.    («  +  $)(a  —  $)  =  a*  —  b. 

§207.   iC2  +  2px  +  !?2=:0  =  (x  —  ai)(x-a2). 

2p  =  —  (a1  +  a2).     q2=aia2. 

x=z-p±J(p*-qS). 
§§  232,  233.   If    a  :  b  =  k  :  I,  then     al  =  bk ; 
§  23d.   a  :  &  r=  5  :  Z ;     ltb  =  k:a; 
§  235.    b  :  a  =  1 :  k; 
§236.   a±6:a  =  £±Z:&; 
§  238.    a  ±  rcJ  :  £  ±  wZ  =  5  ±  ma  :  I  ±  mk  ; 
§  239.   ma  :  nb  ==  mk  :  nl ; 
§241.    an:  bnz=kn:ln. 
§  240.    If  a  :  5  =  e  :/=  g  :  A  =  h  :  2, 

then         a  +  e-[-5r  +  ^  :  ^"H/'+^-M  —  a  :  *• 
§242.    a:b  =  k:l;     e:f—g:h;     r  :  s  —  x  :  y. 
aer  :  bfs  —  hgx  :  Ihy. 


FORMTJLJE.  X1 


§§250,251.   l=a+{n  —  \)D.    *  =  $»(«  + Q- 

a[mn—l)    _bn  —  a 
§§  258-261.   I  =  am""1 ;  *  =  —3-  ~  m~^l ' 

A 

§258.   A=p(l  +  ry.    P=  ^l+ry^ 
(l+r)t—  1        .       ar  1 \ 

And  when  w  =  co, 

^=^1+^+—2-i-r72-3+&c-) 

as  ^  =  ^(2.718  281)*.     §§  330,  342. 
§  273.  No.  ot permutations  of  rc  tilings  =  1.2.3.4 . .  n  —  [»]- 
§  274.  No.  of  arrangements  of  w  tilings,  taken  p  and  ;>  = 
»(n  — 1)  ....  (n—  p  +  l)  =  [«,  w— i»  +  l]- 
$  275.   No.  of  combinations  of  w  things,  taken  ^?  and  jp  = 
?i(?i— 1)  . .  (w— p+l)  _  [w,  n—j?  +  l] 
1 . 2 . 3  .  .  .  p  |>] 

§  280.   If  M  +  JVSc  +  Pa;2  +  &c.  =  0  for  all  values  of  x, 

then  if  =  0 ;  ST=  0  ;  &c/ 

§  294.   (x  +y)n  -  x\  +  je-^  +  ^^-**-23/2  +  &c. 

$295.   i.)  (^±^=^(l±f|+^f^^±&c-) 

,  „„~     ^  .  n(n  —  1) 

§300.   Dn:=±al^:na2±-^- — — ^a3q:&c; 

taking  the  upper  signs,  if  n  is  even ;  and  the  lower, 
if  it  is  odd. 


X"  FORMCLjE. 

§301.  „.  =  a ,  +  (n - 1  )J>,  +  (" ~ *> [(* ~ 2)J8+  Ac. 

§304.  j=  ,.,+sfe^p, +M<"7;1><72 > J,+to 

^307.  ?  =I(± *     ). 

§323.  !^gT=Jf[y_l_|(3,_l)2_j_i(y_i)3_&cj. 

§§  327-8.   J/= —J-  —  U, 

a  —  1—  i(«—  l)2+&c.       Xa 

§§329,330.   J^=  .434  294  481.     e  =  2.718  281. 
§340.  0.=  l  +  Za.*  +  I^:+(Mi|L+&c. 

§351.   a;n4-^41xn~1  .  . -|-^n  =  0  =  (a;  —  aj  .  .  (a;  — a,,;. 

§355.   Ax  =  —  (a1  +a3 +a„); 

i2=a1a2  +  a1fl3-l-&c.     -4„=r±(a1a2  .  .  a„). 

t§ 365-7.    X—sf-^A^-1 .  .+^„:=0,  and#  =  z  —  a/. 

T-y^^-B^-^  ....  +  JB„_1y  +  .£„-0: 

or      r=y:»+ f^*+f%py  +/(*)  =  0. 


The  parenthesis  with  the  sign  of  equality,  it  will  be  ob- 
served, is  sometimes  used  as  an  explanatory  expression. 
Thus  (§  18),  "  10-i(=^)"  is  used  for  "  10"1  (i.  e.  TV-" 


ALGEBRA. 


INTRODUCTION. 

§  1.  Algebra"  is  that  branch  of  the  science  of 
number,  which  employs  general  sy7nbohh  of  quantity*. 

a.)  Arithmetic*,  in  its  largest  sense,  includes  the  whole 
science  of  number  ;  but,  in  its  popular  use,  the  term  is  lim- 
ited to  that  branch  of  the  science,  which  employs  symbols 
of  known  and  particular  numbers  only;  as  2,  3,  10,  12. 

b.)  Algebra,  on  the  other  hand,  employs  general  symbols 
(for  the  most  part,  Italic  letters  of  the  alphabet),  any  one  of 
which  may  represent  any  number  whatever.  Thus  a  rep- 
resents, not  some  particular  number,  but  simply  a  number. 

Note.  Such  symbols  are  termed  algebraic  or  literal",  in  distinc- 
tion from  those  of  common  Arithmetic,  which  are  termed  numerical'. 
A  quantity  expressed  algebraically  is  often  called  an  algebraic  quan- 
tity or  expression. 

c.)  For  convenience  and  perspicuity,  certain  classes  of 
letters  are  usually  appropriated  to  distinct  uses.  Thus,  the 
first  letters  of  the  alphabet,  as  a,  b,  c,  usually  stand  for 
known  or  given  quantities,  and  the   last,  as  x,  y,  z,  for  un- 

(a)  A  word  derived  from  the  Arabic ;  the  Arabs  having  been  among 
the  earliest  cultivators  of  this  science.  (6)  From  the  Greek  av/ij3o?,nr, 
token,  sign,  (c)  From  the  Latin  quantus,  how  much,  (d)  Greek, 
upt&noQ,  member,  (e)  Latin,  littera  or  litera,  a  letter.  (/)  Latin, 
numerus,  number. 

AI,G.  2 


14  INTRODUCTION.  [§  2. 

known  ox  required  quantities;  while  for  exponents  (§  16), 
the  letters  near  the  middle  of  the  alphabet,  as  m,  n,  p,  are 
oftener  used. 

Note.  A  quantity  is  regarded  as  known,  when  it  may  be  assum- 
ed at  pleasure;  as  unknown,  when  it  cannot  be  assumed,  but  must 
be  found  from  its  relation  to  the  known  quantities. 

d.)  A  quantity  is  sometimes  represented  by  the  first  let- 
ter, or  by  several  letters  of  its  name :  thus  interest  is  repre- 
sented by  i ;  sum,  by  S;  difference,  by  D  ;  time,  by  t ;  veloc- 
ity, by  v  ;  radius,  by  r  or  R  ;  sine,  by  sin  ;  cosine,  by  cos  } 
tangent,  by  tan9  ;  &c. 

e.)  Different  quantities  of  the  same  kind,  or  standing  in 
the  same  circumstances,  are  sometimes  represented  by  the 
same  letter  accented.  Thus  similar  known  quantities  may 
be  represented  by  a,  a'  (read  a  prime),  a"  {a  second),  a'"  [a 
third),  &c. ;  similar  unknown  quantities  by  x,  xJ,  x",  &c. 
So,  if  the  radius  of  one  circle  is  represented  by  R,  the  radius 
of  another  may  be  represented  by  R',  &c.  A  distinction  is 
sometimes  made,  by  using  different  forms  of  the  same  let- 
ter ;  as  x,  X;  u,  U ;  r,  R. 

SIGNS. 

§  2.  In  addition  to  the  symbols  of  quantity  above 
mentioned,  Algebra,  in  common  with  other  branches 
of  mathematics,  employs  certain  symbols  of  opera- 
tions and  relations,  called  signs*.     Thus,  the  sign  of 

a.)   Equality,  =,  equal  to  ;  as  1  foot  =  12  inches  ;  a  =  b. 
b.)  Inequality,  1.  Superiority,  >,  greater  than  ;  as  10>7. 
2.  Inferiority,  <,  less  than;  as  7<10;  5«<6a. 

Note.    The  opening  of  the  sign  of  inequality  is  always  towards 
the  greater  quantity. 


(g)  Radius,  sine,  cosine,  and  tangent  are  the  names  of  certain  lines 
;lrawn  in  or  about  a  circle,  and  express  quantities  of  great  import- 
ance, and  of  continual  use  in  the  higher  applications  of  Algebra 
(h)  Latin,  signum,  mark,  sign. 


§  2,  3.]  signs.  lo 

•.)  Addition,  -\-,plus\  or  together  zvith  ;  as  6+4=  10  ;  x-\-a. 
d.)  Subtraction,  — ,  minus',  or  less ;  as  7 — 3  =  4 ;  la — 3a. 

Note.  The  quantities,  which  are  connected  by  the  signs  +  and 
— ,  are  called  terms*. 

e.)  Multiplication,  X,  or . ,  into,  or  multiplied  by;  as  4X5  or 
4.5  =  20;  4aXob  =  20a,b  =  20ab. 

Note.  Between  numbers  and  letters,  and  between  letters  them- 
selves, the  sign  of  multiplication  is  commonly  omitted.  Thus  3abc 
is  the  same  as  3XaX6Xc.  Between  numbers,  on  account  of  the 
local  value  of  figures,  the  sign  can  never  be  omitted.  Thus  35  i< 
not  the  same  as  3X5. 

/".)   Division,  ■—,  divided  by  ;  as  8-^-2  =  4 ;  6a-|-2  =  da. 

Note.  Division  is  more  frequently  denoted  by  writing  the  divi- 
dend above,  and  the  divisor  below  a  fractional  line.    Thus  a  divided 

by  b  is  written-;   8-^-2  — -  =  4. 

g.)  Inference,  .*. ,  therefore,  as  a  =  5,  .\  4a  =  20. 

A.)  Union.  The  parenthesis,  (),  or  vinculum1,  either  hori- 
zontal,   ,  or  vertical,    |  ,  is  used  to  show  that  several 

quantities,  connected  by  the  signs  -4-  or  — ,  are  to  be  tak- 
en together,  or  subjected  to  the  same  operation.     Thus 

(3+4)X2,  or  (3-f-4).2,  or  3+4.2,  or     3  2,  shows  that  3 

and  4  are  to  be  added  together,  and  their  sum  multiplied 
by  2.  So  (a+b)  (a—b)  ;  6— (4— 2)  =  6—2  =  4.  With- 
out the  parenthesis,  the  last  expression  would  be  6 — 4 — 
2  =  0. 

Other  symbols  will  be  introduced  and  explained,  as  they 
are  needed. 

§  3.  It  should  be  remembered  that  these  signs  are  abbre- 
viations for  words  ;  that  they  are,  in  fact,  words  and  phrases 
of  the  algebraic  language. 

(i)  Lat.  plus,  more,  (j)  Lat.  minus,  less,  (k)  Gr.repfia,  bound, 
limit;  Lat.  terminus,  Fr.  terme.     (Z)  Lat.,  tie,  bond. 


16  INTRODUCTION.  [  §  3. 

a.)  Translate  the  following  expressions  into  common 
language. 

a\-b  .  a — b 

L  —  +  —  =«• 

Ans.  The  quantity  obtained  by  adding  b  to  a  and  divid- 
ing the  sum  by  2,  together  with  the  quantity  obtained  by 
subtracting  b  from  a  and  dividing  the  difference  by  2,  is 
equal  to  a. 

Or,  The  half  of  a  plus  b,  plus  the  half  of  a  minus  b,  is 
equal  to  a. 

3.  («+&)  (c-\-x)  =  ac-\-bc-\-ax-\-bx. 

4.  7?Xsin(a+£)=sin  a  cos  6+ cos  a  sin  b.  See  J.  4, 
below. 

5.  aa+aV+a'V+a"'^'^  (a+a'y-fa'")1- 


G.    (100+40+4)12  =  144.  10+2  =  1728,  <200X  10. 

5.)  "Write,  in  algebraic  language,  the  following  sentences. 

1.  10  added  to  4,  and  the  sum  diminished  by  8,  is  equal 
to  3  times  4  divided  by  2. 

Ans.  10+4—8  =  3x4-^-2. 

2.  a  multiplied  by  b,  and  the  product  divided  by  e,  is 
equal  to  x  multiplied  by  a,  and  the  product  diminished  by  b. 

3.  The  diiference  between  a  multiplied  by  x,  and  h  mul- 
tiplied by  y,  is  equal  to  m  multiplied  by  e. 

4.  Radius  into  the  sine  of  the  sum  of  a  and  b  is  equal  to 
the  sine  of  a  into  the  cosine  of  b,  together  with  the  product 
of  the  cosine  of  a  into  the  sine  of  b.     See  a.  4,  above. 

5.  The  sum  of  a  and  b  is  greater  than  c,  and  c  is  greater 
than  the  difference  of  a  and  b. 

The  greater  brevity  and  clearness  of  the  algebraic  lan- 
guage cannot  fail  to  be  observed. 


*4.  ]  POSITIVE  AND  NEGATIVE  QUANTITIES.  17 

POSITIVE  AND  NEGATIVE  QUANTITIES. 

§  4.  In  finding  the  aggregate  of  any  number  of 
quantities,  or  terms  (§2.  d.  N.),  those,  which  tend  to  in- 
crease the  amount,  are  called  positive"1,  and,  as  they 
must  be  added,  are  preceded  by  the  sign  +  ;  those, 
which  tend  to  diminish  the  result,  are  called  nega- 
tive", and  are  preceded  by  the  sign  — ,  to  show  that 
they  must  be  subtracted. 

1.  A  has  Bank  Stock,  to  the  amount  of  $2000,  Real  Es- 
tate, S5000,  other  property,  $1000 ;  he  owes  to  B  $500, 
and  to  C  8300.     What  is  the  net  amount  of  his  property  ? 

Here  the  items  of  property  tend  to  increase  the  amount, 
and  are,  therefore,  positive  ;  the  debts  diminish  the  amount, 
and  are,  therefore,  negative.  The  former  must,  consequent- 
ly, be  preceded,  or  affected  by,  the  sign  +,  and  the  latter, 
by  the  sign  — .  Hence,  we  shall  have,  for  the  true  ex- 
pression of  the  net  value  of  the  estate, 

+2000+5000+1000— 500— 300=  + S7200. 

a.)  The  character  of  every  term  as  positive  or  negath<  . 
must,  of  course,  be  indicated  in  the  expression.  Quantities, 
however,  are  regarded  as  positive,  unless  the  contrary  is 
shown ;  hence,  if  no  sign  is  prefixed  to  a  term,  the  sign  + 
is  always  understood.  Hence,  when  a  positive  term  stands 
alone  or  at  the  beginning  of  a  series  of  terms,  its  sign  is 
usually  omitted.  Thus  5  is  the  same  as  +5 ; 
so  4 — 3  =  +4 — 3  ;  a  =  +a  ;  a-\-b  =  +«+&. 

2.  Let  the  items  of  property  amount  to  $10,000,  the 
debts,  to  $9000.    "What  is  the  aggregate,  or  the  net  estate  ? 

3.  What  is  the  aggregate,  if  the  property  be  represented 
by  a,  and  the  debts  by  b? 

(m)  Lat.  positivus,  from  pono,  to  place,  as  placing  or  giving  value, 
(n)  Lat.  negativus,  from  nego,  to  deny,  as  denying  value. 

2* 


18  INTRODUCTION.  [  §4. 

4.  Again,  suppose  a  surveyor  runs  on  one  side  of  his 
field  20  rods  east,  and,  on  another,  15  rods  west.     What  is 
his  distance  east  of  his  starting  point,  i.  e.  his  departure,  as 
surveyors  call  it?  Am.  20 — 15  =  5  rods, 
or                                E.  20  rods,  W.  15  rods=E.  5  rods. 

The  distance  run  east  is  positive,  because  it  increases  the  distance 
east  of  the  starting  point;  and  the  distance  run  west  is  negative,  be- 
cause it  diminishes  that  distance. 

b.)  As  each  sign  indicates  simply  the  character  of  the 
term  before  which  it  stands,  the  order  of  the  terms  is  obvi- 
ously immaterial,  provided  each  retains  the  proper  sign  be- 
fore it.     Thus  4 — 3  is  the  same  as  — 3+4.     So, 

10—8+6  =  10+0—8  =  6+10—8  =  —8+6+10. 

5.  How  far  will  a  surveyor  be  east  of  his  starting  point, 
if  he  runs  10  rods  west,  and  50  rods  east  ? 

Am.  —10+50  =  50—10  =  40  rods. 

G.  A  owes  $5000,  and  holds  property  to  the  amount  of 
$20,000.     What  is  his  estate  ? 

7.  What,  if  he  owes  a  dollars,  and  holds  property  to  the 
amount  of  b  dollars  ? 

8.  What,  if  he  owes  $5000,  and  holds  $5000  worth  or 
property  ? 

9.  What  is  his  estate,  if  his  property  amounts  to  $5000, 
and  his  debts,  to  $6000  ?     Am.  5000— 6000  =  —$1000. 
or,  property  $5000,  debt  $6000  =  debt  $1000. 

In  this  instance,  $5000  of  the  debt  can  be  paid,  and  there  will  re- 
main $1000  to  be  paid  afterwards,  i.  e.  to  be  subtracted  from  any 
property,  which  may  be  afterwards  acquired. 

10.  A  surveyor  runs  20  rods  east,  and  30  rods  west. 
What  is  his  distance  east  of  his  starting  point  ? 

Am.  — 10  rods, 
or,  E.  20  rods,  W.  30  rods  =  W.  10  rods. 

20  of  the  30  rods  run  west  can  be  subtracted  from  the  20  run  east, 
nnd  10  remain  to  be  subtracted.  Thus,  if  he  should  afterwards  run 
15  rods  east,  his  distance  east  of  his  first  starting  point  would  be 
—10  +15  =  5  rods. 


§4-6.]  POSITIVE  AND  NEGATIVE  QUANTITIES.  19 

c.)  If  it  bud  been  proposed  to  find  his  westerly  distance 
from  the  lirst  point,  the  easterly  distances  would  have  been 
negative,  and  the  westerly,  positive. 

In  like  manner,  if  we  had  proposed,  in  the  examples 
above,  to  find  the  net  indebtedness,  we  must  have  made 
debts  positive,  and  property  negative. 

§  5.  Thus  the  contrary  signs  +  and  —  show  that 
the  quantities,  before  which  they  are  placed,  are  in 
precisely  opposite  circumstances;  that  is,  that  they 
produce  opposite  effects  in  respect  to  the  aggregate 
result ; — that,  as  in  the  case  of  the  distance  east  and 
west,  they  are  reckoned  in  opposite  directions.  In 
other  words,  the  sign  —  is  the  algebraic  expression 
for  contrariwise,  or,  in  reference  to  distances,  back- 
wards. 

Thus,  if  distance  north  be  positive,  distance  south  is  neg- 
ative ;  if,  for  instance,  north  latitude  have  the  sign*-)-,  south 
latitude  must  have  the  sign  — .  If  distance  upward  be 
positive,  distance  downward  is  negative  ;  if  future  time  be 
positive,  past  time  is  negative ;  if  velocity  in  one  direction 
be  positive,  velocity  in  the  opposite  direction  is  negative  ; 
&c. 

§  6.  A  negative  quantity  is  frequently  said  to  be  less  than 
zero.  This  expression  is  most  conveniently  illustrated 
by  examples  8  and  9,  above.  In  example  8,  the  net  estate 
is  0  ;  in  example  9,  it  is  — $1000.  But  a  man,  whose  prop- 
erty is  as  represented  in  example  9,  is  obviously  poorer 
than  he  would  be,  if,  as  in  example  8,  he  were  worth  sim- 
ply nothing.  He  is  worth  less  than  nothing.  It  is  not 
meant,  that  the  thousand  dollars  to  be  subtracted,  is  less 
than  zero ;  but,  that  it  has  less  tendency  to  increase  his 
estate,  than  zero  would  have  ;  that  is,  it  has  a  tendency 
actually  to  diminish  his  estate. 

a.  In  like  manner,  if  he  had  owed  $2000,  he  would  have 
been  worth  less  than  he  is  now,  when  he  owes  only  $1000. 


20  INTRODUCTION.  [§G,  7. 

Hence,  we  say,  that —2000 <— 1000.  That  is,  the  sub- 
traction of  2000  leaves  a  smaller  remainder  than  the  sub- 
traction of  1000.  In  other  words,  — 2000  tends  to  increase 
the  debt  more,  that  is  to  increase  the  property  less,  than 
— 1000,  and  is  therefore  said  to  be  itself  less. 

So,  in  example  5,  — 10  gives  a  greater  distance  west 
and  therefore  a  less  distance  east,  than  — 5  could  have  giv- 
en ;  and  either  of  them,  a  less  distance  east  than  0.    Hence, 
0>—  1;—  2>—  3;  — 5<— 4;  +a  >  0  ;  — a<0. 

b.  Again,  if  we  begin  with  3  and  subtract  1,  we  diminish 
the  amount ;  and  we  continue  to  diminish  it,  as  long  as  we 
continue  to  subtract  1.     Thus, 
3—1  =  2;  2—1  =  1;  1—1  =  0;  0— 1  =—  1 ;  —  1— 1  =—  2. 

Or,  if,  from  the  same  quantity,  we  subtract  continually 
greater  and  greater  quantities,  we  shall  obtain  less  and  less 
remainders.     Thus, 

3—2  =  1;  3—3  =  0;  3— 4  =  — 1;  3— 5  =  —2; 

that  is,  the  greater  the  quantity  to  be  subtracted1  the  less 
the  remainder. 

§  7.  As  a  positive  and  negative  quantity  are  reck- 
oned in  opposite  directions,  the  difference  between 
l hem  is  greater  than  either,  and  is  equal  to  the  sum  of 
the  units  in  both. 

Or,  as  a  negative  quantity  is  less  than  zero,  the  difference 
between  a  positive  and  a  negative  quantity  is  greater  than 
the  difference  between  the  positive  quantity  and  zero ;  and 
greater  by  just  so  much  as  the  negative  quantity  is  less 
than  zero ;  that  is,  by  the  number  of  units  in  the  negative 
quantity. 

1.  A  has  $5000,  and  B  owes  $5000.  What  is  the  dif- 
ference of  their  estates  ?  i.  e.  by  how  much  is  A  richer  than 
B  ?  Ans.  5000+5000  =  $10,000. 

«.)  If  they  should  combine  their  estates,  the  aggregate 
value  would  be  0.     The  difference  between  them  is  clearly 


§  7,  8.]        POSITIVE  AND  NEGATIVE  QUANTITIES.  21 

$10,000,  the  sum  which  B  must  obtain,  in  order  to  be  as 
rich  as  A.  This  difference  is  expressed  thus,  5000 — 
(—5000).     Hence, 

5000— (—5000)  =  5000+5000 ;  or  —(—5000)  =+5000. 
So  — ( — a)  =  +a.     Hence, 
b.)    The  subtraction  of  a  negative  quantity  has  the  same 
effect  as  the  addition  of  an  equal  positive  quantity. 

2.  The  latitude  of  New  Orleans  is  30°  North;  that  of 
Buenos  Ayres  is  34°  South.  How  many  degrees  is  the 
one  place  North  of  the  other  ?  That  is,  what  is  the  differ- 
ence of  their  latitudes  ? 

3.  X  has  a  dollars,  and  Y  owes  b  dollars.  "What  is  the 
difference  between  their  estates  ? 

Ans.  a — ( — b)  ~a-{-b,  as  in  example  1. 

4.  At  sunrise  on  the  20th  of  February,  the  thermome- 
ter stood  at  30°  below  zero;  at  sunrise  on  .the  20th  of 
March,  it  stood  at  30°  above  zero.  What  is  the  difference 
in  the  temperatures  ? 

5.  The  reading  of  the  thermometer  on  one  day  is  — 10° 
(10°  below  0)  ;  on  another  day,  it  is  — 20°.  Which  indi- 
cates the  greater  heat  ?     How  much?     §G.  a  and  b. 

§  8.  The  process  of  finding  the  aggregate  of  several  quan- 
tities, regard  being  had  to  their  character  as  positive  or 
negative,  is  algebraic  addition  ;  the  process  of  finding  the 
difference  between  quantities  so  considered  is  algebraic  sub- 
traction. Arithmetical  addition  and  subtraction,  on  the 
other  hand,  relate  to  numbers  regarded  simply  as  such, 
without  distinguishing  them  as  positive  and  negative. 

(a)  The  algebraic  sum  may  be  less  than  the  algebraic 
difference  (§7.  a)  ;  and  (b)  the  algebraic  sum  may  be  equal 
to   the  arithmetical  difference  (§4)  ;   or  (c)   the  algebra!' 

'Ference,  to  the  arithmetical  sum. 


22  INTRODUCTION. 


FACTORS  AND  POTTERS. 

§  9.  Quantities  multiplied  together  are  called,  a? 
in  Arithmetic,  factors"  in  respect  to  the  product. 
and  are  also  called  coefficients'2  in  respect  to  each 
other. 

Thus,  in  the  expressions  3a,  2a,  ba,  bca,  and  ha,  3,  2,  b, 
be  and  I  are  coefficients  of  a.  In  3xy,  3  is  the  coefficient 
of  xy ;  ox.  of  y  ;  and  3y,  of  x. 

a.)   The  coefficient  shows,  how  many  times  the  quantity 
multiplied  is  taken  as  a  term  (i2.  d.  N).     If  the  coefficient 
is  positive,  it  shows  how  many  times  the  quantity  is  added : 
if  negative,  how  many  times  it  is  subtracted  (§4).     Thu-. 
3a  =■  a-\-a-\-a  :  2x  =.  x-\~   . 
— 3X+a  —  — a — a — a  =  3X — a  =  — 3a. 
So  —aX-\-b  =  aX—b  =  —ab. 
—2X—a  =  —(—a)—(—o )  =  a+a  (§  7.  a,  b)  =  2a. 

Note.  In  the  last  example,  — a  is  to  be  subtracted  twice;  and 
subtracting  — a  twice  has  the  same  effect  as  adding  +a  twice  (§7  b). 

Hence,  if  two  factors  multiplied  together  are  both  posi- 
tive or  both  negative,  the  product  is  positive  :  if  one  is  posi- 
tive and  the  other  negative,  the  product  is  negative.  Or. 
more  briefly. 

Like  signs  give  -{-,  unlike,  — . 

1 .  What  is  the  product  of  2a  and  — b  ?    of  — 2ab  and  — c  ? 

aX— xy  =  what?     —3aX—.ry?     —3aX—xy?    —2 
X— 3  ? 

h.)  A  letter,  or  combination  of  letters,  used  as  a  coeffi- 
cient, is  called  a  literal  coefficient ;  a  number,  so  employed, 
is  called  a  numerical  coefficient.  Coefficients  are  also  dis- 
tinguished as  integral  or  fractional,  &c. 

(o)  hut.,  maker, producer,  (p)  L.  productus,  produced,  i  e.  by 
the  multiplication.  (</)  Lat.  coefficio.  to  aid  in  forming,  a  co-fac- 
tor,    (r)  Lat.  integer,  ivhole  :  numbers  are  called  integral  or  whole. 


5  0,   10.]  FACTORS  AND  POWERS.  % 

When  no  numerical  coefficient  is  expressed,  1  is  always 
implied.     Thus  a  is  the  same  as  la;  x=lx;  abzzzlab. 

1.  In  labcx,  wh  t  is  the  coefficient  of  x?  of  ex?  of  bcx'i 

2.  In  xyz,  what  is  the  coefficient  of  x?  ofy?  of  xyz? 

3.  In  (a-\-b)(a — b)c,  what  is  the  coefficient  of  c ?  of  a-\-b  ? 
of  a— J  ? 

4.  In — 5  a  far,  what  is  the  coefficient  of  a:?  of  ox?  of 
— 5x  ?  — 5a  ?  — ab  ?  abx  ?  — abx  ? 

§  10.  The  combining  of  factors  into  a  product  is  the 
work  of  multiplication :  the  separation  of  a  given  fac- 
tor from  a  given  product  is  the  work  of  division. 

Thus,  by  multiplication,  we  combine  the  factors,  3  and  4. 
into  a  product  12  :  by  division,  we  separate  the  given  fac- 
tor 3,  from  the  given  product  12,  and  find  the  other  factor 
4. 

a.)  The  given  product  is  called,  in  reference  to  division, 
the  dividend'' ;  the  given  factor  the  divisor1 ;  and  the  re- 
quired factor,  the  quotient". 

b.)  The  divisor  and  quotient  are  the  factors  of  the  divi- 
dend. They  are,  therefore,  coefficients  of  each  other.  If 
then  the  letters  of  the  divisor  be  found  in  the  dividend,  we 
have  only  to  suppress  or  cancel  them,  and  the  remaining 
factors  constitute  the  quotient  (division  of  the  numerical  co- 
efficients being  performed  as  in  Arithmetic). 

Thus  ab-^b  —  a  ;  abx-^-ab  zzz  x ;  7  abcxy-^T  ac  =zl  bxg. 

1.  2abx-r-b  =. what ?  IQabcxyz-^Sabz? 

2.  3.4.5.6-^3.6  =  what?  1.2.3.4.5.6^6.5.4? 

3.  o-r<*=what?  ab+ab?  1.2.3-^1-2.3? 

Note.    When  the  divisor  is  equal  to  the  dividend,  the  quotient  is 
ob%iously  unity. 

in  distinction  from  fractional  (Lat.  frango,  to  break),  or  broken 
numbers,  (s)  Lat.  Dividendus,  to  be  divided.  (I)  Lat.  divisor,  a 
divider  from  divido,  to  divide,  or  separate,  (u)  Lat.  quoties,  or 
quotiens,  how  many  times,  as  it  shows  how  many  times  the  divisor  is 
contained  in  the  dividend. 


24  INTRODUCTION.  [§10. 

c.)  If  the  divisor  contain  factors,  which  are  not  found  in 
the  dividend,  we  may  cancel  the  common  factors,  and  ex- 
press the  division  by  the  remaining  factors  of  the  divisor  in 
the  usual  form  (§2. /and  ~N). 

2bc 

Thus  lahc-^ax  =  2bc-Jrx,  or . 

x 

labxu         ,      „     abx  _     Ibex  „     sin  a  cos  &  „ 

1.  -  =  what? ?     ? ? 

bcxr  be  3bc         cos  a  cos  b 

a    1.2.3.4        .      .     20.     5.4.3.2.1  „ 

2. =  what  ?     -r 


4.5.  15  1.2.3 

Note.  This,  it  will  be  observed,  is  equivalent  td  the  process  of 
reducing  a  fraction  to  its  lowest  terms.  This  process  may  be  applied 
in  all  cases.  Whenever  all  the  factors  can  be  cancelled  out  of  either 
the  divisor  or  the  dividend,  unity  will  be  found  in  their  place.  If 
this  happen  to  the  divisor,  the  quotient  will  be  found  in  the  usual 
form  as  above  (b)  ;  if  to  the  dividend,  unity  will  stand  above  the  line, 
or  in  the  place  of  the  dividend,  and  the  remaining  factors  of  the  divi- 
sor will  stand  below  the  line,  or  after  the  sign;  if  to  both  divisor  and 
dividend,  the  result  will  be  1-^-  1  =z  1. 

d.)  If  the  dividend  is  positive,  its  factors  (the  divisor  and 
quotient)  must  have  like  signs  (both  positive,  oi:  both  neg- 
ative) ;  and  if  the  dividend  is  negative,  its  factors  must  have 
unlike  signs  (one  positive,  and  the  other  negative)  (See 
§  9.  a).    Therefore. 

If  the  dividend  is  positive,  a  positive  divisor  gives  a  posi- 
tive quotient;  a.  negative  divisor,  a  negative  quotient;  if  the 
dividend  is  negative,  a  positive  divisor  gives  a  negative  quo- 
tient;  a  negative  divisor,  a  positive  quotient.  Hence,  as  in 
multiplication, 

Like  signs  give  -f-,  unlike,  — . 

Thus,  ^-  =  +b,  for  (+a)(Jrb)  =  +abi   ±^  =  +4. 
-\-b,foT(~-a)(+b)  =  —ab;  ' 


-a 


—3 


A-ab  -4-12 

TUL  =  -b,  for  (-a)  (-b)  =  +ab ;  ^  =  -4. 


§  11.]  FACTORS  AND  rOWEKS.  25 

=£  =  -&,  for  (+«)  (-J)  =  -*>& ;  ~  =  -4. 

1.  _9«Z,_^_2a;=what?     —  2ab-+2a?     2ax^-—a? 

2.  _iOx-| 10  =  what?     60-; 10?     —  GO-; 10? 

§  11.  When  a  factor  occurs  more  than  once  in  a 
product,  it  is  usually  ivritten  but  once,  and  the  num- 
ber of  times  it  is  employed,  is  denoted  by  a  number 
or  letter  placed  over  it  at  the  right,  called  an  expo- 
nent", or  index™. 

Thus,  instead  of  aa,  aaa,  bbbbbb,  we  write  a2,  a3,  bG  ; 
instead  of  2.2,  2.2.2.2,  3.3.3.3.3.3,  we  write  22,  24,  3e, 
the  exponent,  in  every  case,  showing  how  many  times  the 
quantity  over  which  it  is  placed 'is  taken  as  a  factor;  in 
other  words,  how  many  equal  factors  the  product  contains. 
Thus,  in  the  expression,  (a~\-b)3,  the  exponent  3  shows 
that  a-\-b  is  taken  three  times  as  a  factor,  or  that  the  pro- 
duct consists  of  three  factors  each  equal  to  a~\-b.  So,  the 
product  a'2b3x6  contains  two  factors  equal  to  a,  three  equal 
to  b,  and  five  equal  to  x. 

1.  Write  2.2.3.2.3.2  with  exponents.  Am.  24.32. 

2.  Write  aabcabac  with  exponents. 

3.  Write  23.\03.o±.b*  without  exponents. 

4.  Write  aib3c2x5y6  without  exponents. 

Note  1.  These  expressions  may  be  read  thus;  a2, a  taken  twice  as 
a  factor;  b3,  b  taken  three  times  as  a  factor;  &c.    Also,  a1,  (§11. 

a),  ao  (§13),  a  taken  once,  a  taken  no  times  as  a  factor;  a2" 
(§  12)  a  taken  half  a  time  as  a  factor;  a~2  (§  14),  a  taken  minus 
twice  as  a  factor;  &c.  Or,  if  the  teacher  prefer,  the  student  may 
examine  §  22  and  a  under  it,  and  use  the  expressions  given  there. 

Note  2.  A  negative  quantity  may  obviously  occur  more  than  once 
as  a  factor;  as  (—  a)(—a)  =  (—  a)2  ;  (—  b)(— 6)(— b)  =(—  b)3. 
In  such  cases,  if  the  number  of  factors  be  even,  the  product  will  be 
positive;  for,  if  they  be  combined  two  and  two,  the  product  of  each 


(t>)  Lat.  exponens,  setting  forth,  showing,     (w)  Lat.  indicator' 
mark. 


ALG. 


26  INTRODUCTION.  [§12. 

pair  will  be  positive  (§  9.  a) ;  and  the  product  of  these  positive  pro- 
ducts will,  of  course,  be  positive.  If  the  number  of  factors  be  odd, 
the  greatest  even  number  will  give  a  positive  product,  and  this,  mul- 
tiplied by  the  remaining  negative  factor,  will  give  a  negative  pro- 
duct (§9.  a).    Hence, 

If  the  number  of  negative  factors  be  even,  the  product  will  be  pos- 
itive; if  odd,  negative.     Thus, 

(— a)a=+a*;   (— a)3  —  —a3; 

(— a;)*  =  -f-a?*;   (— x)*  —  —x5. 

a.)  When  a  quantity  is  taken  as  a  factor  only  once,  the 
fact  may  be  shown  by  the  exponent l ;  but  in  this  case,  the 
exponent  is  usually  not  written ;  and  whenever  no  exponent 
is  written,  1  is  always  implied.  Thus  a  is  the  same  as  o1  j 
ax=a1x1  ;  ax2  =za1x2. 

§  12.  b.)  The  fraction  J  shows  that  the  unit  is  separated 
into  two  equal  parts,  and  that  only  one  of  them  is  taken. 

i  i 

So  the  exponent  2,  in  the  expression  a2,  shows  that  a  is 

separated  into  two  equal  factors,  and  that  only  one  of  them 
is  employed ;  in  other  words,  that  a  is  introduced  as  a  fac- 
tor, half  a  time.     If  this  half-factor  were  introduced  two, 

2.      a     ± 
three,  or  four  times,  we  should  have  a2,  a2,  a'-.     Thus, 

£  i.    x     Li  i  4 

a2  =  a2.a2.a~.a2  =  (a2)  . 

If  a  were  separated  into  three,  four,  or  n  equal  factors, 

11-. 

and  one  only  employed,  we  should  write  a3,  a*,  «"  ;    if  tioo 


were  employed,  a5,  a4,  an ;  &c.     Hence, 

The  denominator  of  a  fractional  exponent  shows,  into  how 
many  equal  factors  the  quantity  under  the  exponent  is  sepa- 
rated ;  and  the  numerator  shows,  how  many  of  these  fac- 
tors are  employed.     Thus, 

9^  =  3  ;  9^  =  9^9^=3.3  =  9;  9  *  =  3.8.3  =  27. 

8^  =  2;  8^  =  2.2  =  4;  8^  =  2.2.2.2  =  16. 

1 .  What  is  the  meaning  of  (aa)  2  ?     of  i? 2  ?     of  a$  ? 

2.  (#2)*  =  what?     16*!     27*?     25^?     36*?     49^? 


§  13.]  FACTORS  AND  POWERS.  27 

e.)  Otherwise,  as  f  =  i  of  4,  a2  indicates,  that  one  half 
of  four  factors  each  equal  to  a  are  introduced  ;  or  that  a 
had  been  introduced  four  times  as  a  factor,  and  the  pro- 
duct, so  formed,  had  been  afterwards  separated  into  two 
equal  factors,  of  which  only  one  was  actually  employed. 
Thus, 

a  $  =  a*X  2  =  (a4)  *=  (aaaa)  ?=zaa=az. 

Hence,  again, 

The  numerator  of  a  fractional  exponent  shows,  how  ma- 
ny times  the  quantity  under  the  exponent  has  been  em- 
ployed as  a  factor ;  and  the  denominator  shows,  into  how 
many  equal  factors  the  product  so  formed  has  been  separated. 

1         £        3       ™ 

Thus  a3,  a3,  a4,  an  indicate,  that  a,  a2,  a3,  am  have  been 
separated,  the  first  two  into  3,  the  third  into  4,  and  the 
fourth  into  n  equal  factors,  of  which  only  one  is  employed. 
Or  that  a  is  employed  as  a  factor  £,  ~,  f ,  "  of  a  time. 

Thus,  9^  =  (9 2)2"  =  (9.9)'  =  9;  8^=  (82)^=  (8.8)^ 
=  64*=  (4.4.4)*=  4. 

(i?2)5—  (R2.R2.Esy=(B.B.B.R.R.li.)i=(RG)?= 

1.  What  is  the  meaning  of  (aa)*?  of  22?  of  3-? 
of32?    ofxh 

2.  (£2) 2  —  what?     16*?     27*?     5*?     22?     (x3)*? 

§  13.  d.)  Any  quantity,  which  is  not  found  as  a  factor  in 
a  product,  may  be  introduced  with  zero  for  an  exponent. 
For  this  exponent  will  show,  that  the  quantity,  though  writ- 
ten, still  is  not  employed,  or  is  employed  no  times,  as  a  fac- 
tor ;  and,  of  course,  the  value  of  the  expression  is  the  same 
as  if  the  quantity  were  not  written.  Thus  a°bx2  is  the 
same  as  bz2 ;  ax°  =  a;  but  oXl=a;  that  is,  aX#°  = 
aXl;  .-.  x°  =  l.     Hence, 


28  INTRODUCTION.  [§  14. 

Corollary  I.  Any  quantity  with  zero  for  its  exponent  is 
equal  to  unity. 

Note.  A  corollary1  is  an  inference  from  a  preceding  prin- 
ciple. 

§14.  e.)  "When  a  factor  is  introduced  less  than  no  times 
(§  6.),  i.  e.  -when  instead  of  being  introduced,  it  is  taken  out, 
the  fact  will  be  properly  indicated  by  a  negative  exponent 
(§§4,5).  But  a  factor  is  taken  out  by  division  (§10). 
Consequently,  a  negative  exponent  shows,  that  the  quantity 
under  it  is  to  be  employed  as  a  divisor,  as  many  times 
(§  11),  or  parts  of  a  time  (§  12.  b,  c),  as  there  are  units  or 
parts  of  a  unit  in  the  exponent.  Thus,  in  the  expression 
a~1x,  a,  instead  of  being  multiplied  into,  is  to  be  divided 

x 

out  of  x,  and  the  expression  is  therefore  equivalent  to  -. 

a 

Also,  in  the  expression  a  2x,  the  negative  factional  ex- 

ponent  2  indicates,  that  a  is  separated  into  two  equal  fac- 
tors, and  that  one  of  these  half-factors  (§  12.  b)  is  taken  out 
five  times  by  division  ;  i.  e.  that  the  whole  factor  a  is  taken 
out  five  halves  of  a  time.  This  is  evidently  the  same  thing 
as  saying,  that  it  is  introduced  minus  five  halves  of  a  time. 

In  other  word3  a  2  indicates,  that  a  product,  containing  a 
five  times  as  a  factor,  is  separated  into  two  equal  factors, 
and  that  one  of  these  two  factors  is  to  be  taken  out  by  di- 

vision.      The   expression  is,  therefore   equivalent  to  — . 


a" 


So, 


*-»*-*.  10-112-^-  2-  5'-ii-^- 

9-*.6  =  -^  =  |  =  2.     See§§17,  19. 
^      3 
1.  2~1.3  =  what?     3-V2?     10~2.30?     I5-1.80? 


(x)  Lat.  corollarium,  something  given  over  and  above,  from  co- 
rolla, a  wreath,  a  common  present  or  mark  of  honer. 


§  15,  16.]  FACTORS  AND  POWERS.  29 

2.  «-35  =  what?   a°b~lx?     b~*x?    aV^ar1?   a~mxm? 

§  15.  /.)  If  a  quantity  be  found  any  number  of  times  in 
n  multiplier  and  multiplicand,  it  will  be  found  in  the  pro- 
duct as  many  times  as  in  both  the  factors.     For 

a2b3Xaib=zaabbbXaaaab  =  aaaaaabbbb  (§2.   e.  N)  = 

aGb*.     Hence, 

Cor.  II.  The  exponent  of  any  quantity  in  a  product  wil 
be  equal  to  the  sum  of  its  exponents  in  the  factors. 

1.  a2  b2Xab  =  what?  ax2Xa2x?  a°bc*Xa*bc*  ?  a3 
x°X«3y? 

2.  23.33X2.32=what? 

Ans.  2*.35  =  16X243  =  3888. 

3.  22.34x2°.3  =  what?     53.2x5°2?     102X103  ? 

4.  100*X  100^  =  what? 

Ans.  10.103  =  10*  =  10,000. 

5.  100*X100  =  what?    25*25*?    27*.  271?    16*.16*? 

6.  a*Xa2  =  what?   cfix<$?   162.162?    10*.  10$?. 

§  16.  g.)  It  is  also  evident,  that  the  exponent  of  a  quan- 
tity in  one  of  the  factors  must  be  equal  to  the  exponent  of 
that  quantity  in  the  product,  minus  its  exponent  in  the  oth- 
er factor.     Hence, 

Cor.  III.  The  exponent  of  any  quantity  in  a  quotient  is 
equal  to  the  exponent  of  that  quantity  in  the  dividend,  mi- 
nus its  exponent  in  the  divisor. 

5. 

m       a5      aaaaa  n     103      „.„      a2         4 

Thus—  = —  aa  —  a2 ;  -— =102;    —  =  a*  = 

a3        aaa  103 


a 


{aaaaf  =  a2 


,    x3         ,      _     a7b3  r     «2J2       a3b2  .     ax5 . 

1.     — -  =  Whftt?  rr—i         —  ?         5 ? ? 

x2  a*b  ab  ab2         a°x 


x 


3 


2.  —  =  what ?     ^4w5.  a;3-3  =:x°  =  1  (Cor.  I). 

*3 


30  INTRODUCTION.  [§  17,  18 

3.  cfi-r<$  —  what?    102-M0*?    a+c$?    a^~a2? 

4.—-  =  what?     Ans.  x3~ *  =  x~ 1,  or  — =z—.    Hence 
x*  x±      x 


§  17.  It.)  We  have  ar-i  =-.     See  §  14 


x 


-/,«?      '/*'>     IT10  Y* 

In  like  manner,  — ,  — ,  — ,  -— ,  give  x~2,  x~3,  x~*, 

1111 

«— *  =  — ,  — ,  — -,  — -,  respectively.     Hence, 
x2' x3' x*  x11'       l  J  ' 

Cor.  IV.  A  quantity  xoith  a  negative  exponent  is  equal  to 

unity  divided  hy  the  same  quantity  with  an  equal  positive 

exponent. 

§  18.  ?'.)  The  quotient  obtained  by  dividing  unity  by  any 

quantity  is  calied  the  reciprocal1'  of  that  quantity.    Thus 

1    1  1       1  -o- 

-,  — ,  1-f-a,  — -,  -7-T,  1-i-a2,  or  the  equivalent  expressions 

x  x2  10    10" 

_i 
x— l,  x—-,  a-1,  10—  x,  10— 2,  a  2,are  the  reciprocals  of  x, 

x2,  a,  10,  102  and  a-  respectively.     Also  the  reciprocal  of 
10~1(=  TV)    is   jrpr  =  1  -T-  tV  =  10  5*  tlie  reciprocal  of 

a-2  (=\)  is  -i^l-r-  4  =  «2-*     Hence, 
•     \     a2/       a —  a- 

To  express  the  reciprocal  of  any  quantity,  we  have  only 
to  change  the  sign  of  its  exponent. 

Write  the  answers  to  the  following  questions  both  by  means  of 
exponents  with  their  signs  changed,  and  under  the  fractional  form. 

What  is  the  reciprocal  of  2  ?  of  3  ?  of  10  ?   of  J  ?   of  \  ? 

ofTV(=10-i)?    ofi?    of.01?     ofx2?     ofar-3?     0f 

0-3?     of92?     of  8^  ?     of  25"^  ? 

(y)  Lat.  reciprocus,  returning  upon  itself,  mutual. 

*Note.  This  is  evidently  true;  for,  if  a  unit  be  divided  into  10 
equal  parts,  one  of  them  will  be  contained  in  any  quantity  10  times 
as  often  as  the  whole  unit  is  contained  in  the  same  quantity;  and,  if 
thi  unit  be  divided  into  a2  equal  parts,  one  of  these  parts  will  be 
contained  a2  times  as  often  as  the  whole  umt. 


§10.]  FACTORS  AND  POWERS.  3i 

L)  Positive  and  negative  exponents  have  the  Bame  rela- 
tion of  contrariety  or  oppositeness  as  other  positive  and 
negative  quantities  (§5).  Thus,  an  exponent  shows,  how- 
many  times  or  parts  of  a  time  a  quantity  is  introduced  as  a 
factor.  The  opposite  to  introducing  a  factor  is  taking  it 
out.  When  therefore  a  quantity  is  said  to  be  introduced 
minus  three,  or  minus  n  times,  as  a  factor,  it  is  the  same 
thing  as  saying  that  it  must  be  taken  out  three,  or  n  times 
(§14).  Thus,  in  example  fourth  (§  16),  x  can  be  taken 
out  three  times,  and  the  fact,  that  it  is  to  be  taken  out  once 
more,  is  indicated  by  the  negative  exponent  — 1  (§  4.  c). 
It  is  to  be  so  taken  out,  whenever,  in  subsequent  multipli- 
catiun,  x  shall  be  introduced. 

If  the  operation  be  represented  and  performed  in  the 

x3       1 
fractional  form,  we  have  —  =  — ;  that  is,  three  of  the  four 

x+       x 

factors  of  the  divisor  are  cancelled  out  of  the  dividend,  and 
one  remains  to  be  taken  out,  whenever  x  shall  be  introduced 
into  the  dividend. 

§  19.  I.)  As  the  factors  under  the  negative  exponent  di- 
minish the  whole  number  of  factors  in  the  product,  there- 
fore, 

(1.)  In  combining  the  exponents  of  a  letter  in  the  factors, 
to  find  its  exponent  in  the  product,  the  negative  exponents 
must  be  treated  precisely  as  negative  terms  in  making 
ud  an  aggregate  (§4).     Thus  a3X«—1  =a2  ;    x5Xxr~ 4,= 

xy-  —X. 

1.  x8X^-6=what?     x13Xar4?     a2b2Xa~1b? 

2.  x2Xx~~ 5  =  what?  Ans.  x2~ 5  =  x~ 3. 

x2 
But  — —  x2~  5—x~  3. 

xb 

.    .  i*/        /\  \Aj  — —  *Aj  ^    *A*       . 

(2.)  In  like  manner,  if  negative  exponents  be  found  in 
a  divisor  or  dividend,  they  must  be  treated  like  negative 
terms  in  finding  a  difference  (§7).     Thus, 


3S  INTRODUCTION.  [§20,21. 

a*-^a-3  =  a2-  (-3)=  a5.       See  §  7.  a,  h. 
But  a2Xa3=fl5. 

a2-ra-3  =  a2Xa3. 
Hence  (1,  2), 

Cor.  IV.   To  multiply  or  divide  by  a  quantity  with  a  neg- 
ative exponent,  is  the  same  as  to  divide  or  multiply  by  the 
quantity  with  an  equal  positive  exponent. 
Or,  more  generally, 

To  multiply  or  divide  by  any  quantity  is  the  same  at 
to  divide  or  multiply  by  its  reciprocal. 

Thus  ab2cx2y-^-abc  =  ab2cx2y  X  a~1b-1c~i  =  bx*y. 
22.3  -±  2.3  =  22.3  X  2-i.3-i  =  2. 

1.  abc  -f-  a  be  z=  what  ?  abcXa—1b~1~c1?  o2xX«-1x? 
a-x~ax~ l  ? 

2.  22.3 2.42-^-2.3.4rr: what?  22.32.42-^2-1.3~i.4-1  ? 
2.3.10-^2.3  ?     2.3.10-^2- 1.8-1  ? 

3.  cAcf*  =  what?    «_2\a~2?    10-2\10~*  ?    x.x~h 

4.  a2-^-a~2"  =  what  ?  z-^""^  ?  cT^+cT^?  10^-10"*? 

§  20.  m.)  When  a  quantity  is  taken  a9  a  factor  any  num- 
ber of  times,  and  the  product  so  formed  is  again  taken  as  a 
factor  any  number  of  times,  the  first  quantity  will  evident- 
ly be  employed  a  number  of  times  equal  to  the  product  of 
the  exponents.     (See  §  12.  Examples.)     Thus, 

(a3)3=a3.a3.a3=a<>;  (a^  =  flW  =  o^; 

1.  («-4)3=what?  (23)2?  (102)*?  (x*y?  (as)~h 
(am)n?     (2  s)3? 

?i.)  Thus  we  see  that  the  exponent  may  be  either  inte- 
gral or  fractional,  positive  or  negative,  and  it  may  be  either 
known  or  unknoion. 

§21.  0.)  The  analogy,  as  well  as  the  difference,  between 
the  coefficient  (§  9.  a)  and  exponent,  is  very  obvious.  Both 
relate  to  the  introduction  of  equal  quantities ;   the  coeffi- 


§  21.]  FACTORS  AND  POWEKS.  33 

dent,  of  equal  terms  (§  2.  d.  K)  ;  the  exponent,  of  equal /ac- 
tors (§  9).  If  positive,  they  ajw-m  the  introduction  of  the 
quantities;  the  coefficiently  addition  (§  9.  a);  the  exponent, 
by  multiplication  (§  11).  If  negative,  they  aVfty  the  intro- 
duction, i.  e.  they  affirm  the  removal  or  taking  out  of  the 
quantities  ;  the  coefficient,  by  subtraction  (§  9.  «)  ;  the  ex- 
ponent by  division  (§  14).  If  fractional,  they  show  the  in- 
troduction or  removal,  by  addition  or  subtraction,  or  by 
multiplication  or  division,  as  the  case  may  be ;  the  coeffi- 
cient of  equal  fractional  parts  (§  9.  J)  ;  the  exponent,  of 
«0M«£  components"  (§  12)  of  the  quantity. 

That  is,  they  show,  how  many  times  or  parts  of  a  time, 
a  quantity  is  introduced  or  taken  out ;  the  coefficient,  as  a 
term  ;  the  exponent,  as  a.  factor.  In  other  words,  they  show 
the  introduction, positively  or  negatively  (§  4),  of  a  term  or 
factor,  so  many  times  as  there  are  units  in  the  coefficient  or 
exponent. 

Thus  +2X4  =  4+4  =  +8;   4+2  =  4x4=16. 

_2X4  =  -4-4  =  -8;    4-«  =  ^=Q=^ 

+£.16  =  i(S+8)  =  8 ;    16+*  =  (4.4)*  =  4. 

_i.l6-i(_8-8)=-8;  16"*  =  -^-  =  —^  =  \ 

16*      (4X4)2      * 

So,         x-{-0Xa=.xJr0  =  x;  xXa°—xXl—^- 

1.  Write  abbbccxxxx  with  exponents. 

^4«s.  a1&3c2x4. 

2.  Write  in  like  manner,  aayy,  fefcc,  (a-\-b)(a-\-b). 

3.  Write  a-b3x1y°  without  exponents.     Ans.  aabbbx. 

4.  Write  in  like  manner,  a4j;5,  (a+6)2,  (a — 6)3. 

5.  Write  with  exponents  2X2x3x2x3x4. 

Ans.  23.32.41. 

6.  Write  with  exponents  2X2X2X3X3X2X3X4X4. 

7.  43  — what?  Am.  4X4x4=64. 

(z)   Lat.  coinpono,  to  compose  ;  factors,  which,  multiplied  together, 
produce  a  quantity,  are  called  its  components. 


INTRODUCTION.  [§  22. 

8.  4*  =  what?    53?     72?     10*?     105?     212? 

9.  What  is  the  difference  between  10x4  and  104? 

10.  Show  the  difference  between  3a  and  a3  ? 

Arts.  3a  —  a-{-a-\-a ;  a3=zaXaXa. 

11.  What  is  the  difference  between  a°  and  aXO?   be- 

•ween  10°  and  10X0?   between   1°  and  1X0?   between 

■»u,  1°  and  10°  ?    between  1°,  l1  and  l2  ? 

i 

12.  Show  the  difference  between  \a  and  a2. 

Ans.  \a-\-la  =  a,  crXa   =  a. 

i 

13.  What  is  the  difference  between  l00XHand  1002  ? 

ween  i  of  9  and  9-  ?    between  £.27  and  27 3  ?    between 

§.16  and  16^? 

14.  What  is  the  difference  between  — 32,  3~2,  3—2  and 
3(-2)? 

^s.-32=-9;  3~2=p  =  i;  3-2  =  1 ;  3(-2)  =  -6. 

15.  Write  in  like  manner  6,  8,  10  and  15,  and  interpret 
the  exnressions. 

1G.  What  is  the  difference  between  9"   and  9  2?     83 

and  8~f  ? 

17.  What  is  the  reciprocal  of  10  ?     of  102?     of  100? 

of— 10?     ofl?     of  a?     of-^r?     of-?     of  a-1?     ofa?? 

10  a 

'«-"?     of  100*?     of  27"*?     of  8^? 

18.  a2-^a=what?     a*+a°?     a*+a2?     a2-~a3? 

19.  «3-^-«  =  what?     o3H-a°?     a3-^"1  ?     a3-H*-2? 

20.  Substitute  10  for  a  in  the  last  two  examples. 

22.  If  a  is  employed  m  times,  and  b,  n  times,  what  is  the 
expression  for  their  product  ? 

§  22.    Any  quantity  ivith  an  exponent,  is  called  a 
i'ower  of  the  quantity  under  the  exponent. 


§  22.]  FACTORS  AND  POWERS.  35 

Note.    The  quantity  under  the  exponent  is  called  the  base,,  of 
the  power. 

a.)  A  power  is  designated  by  its  exponent.     Thus, 
x~2  is  read  x  minus  second  power;  x~1,  x  minus  first  power. 
'"         "     a:  zero  power;  x1,  x  first  power. 
x2         "     x  second  power  or  square* ;  re3,  x  third  power  or 
cube0. 

1  2. 

Xs         "    x  one  half  power  ;  x3,  x  two  thirds  power. 

_i 
x  2       "    x  minus  one  half  power,  &c. 

b,)  It  will  be  observed,  that  the  term  power,  as  used 
here,  has  a  wider  signification  than  is  attached  to  it  in 
Arithmetic.  In  Arithmetic,  the  term  is  applied  only  to  a 
product  of  equal  factors.  As  here  defined,  it  includes  a 
single  factor  (§  11.  a),  unity  (equal  to  the  zero  power  (§  18) 
of  a  factor),  and  all  products  and  quotients  formed  by  mul- 
tiplying and  dividing  (§  14,  17)  unity,  any  number  of  times, 
by  the  factor,  or  by  any  of  its  equal  components  (§  12, 14). 

c.)  We  have  therefore  several  classes  of  powers,  distin- 
guished by  the  characters  of  their  exponents.  Thus,  there 
are 

(1.)  Powers  with  positive  integral  exponents  (§  11),  the 
same  as  ordinary  arithmetical  powers  ; 

(2.)  Powers  with  positive  fractional  exponents  (§12), 
consisting  of  equal  components  and  their  combinations  ; 

(3.)  Powers  with  negative  integral  exponents  (§  14),  the 
reciprocals  of  the  first  class  ;  and 

(4.)  Powers  with  negative  fractional  exponents  (§14), 
the  reciprocals  of  the  second  class. 

d.)  Powers  of  these  several  classes  are  sometimes  called 
positive,  negative,  &c,  powers  ;  meaning,  not  that  they  are 
positive  or  negative,  integral  or  fractional  quantities,  but 

(a)  Gr.  /3aaif,  foundation,  (b)  Lat.  quadra,  Fr.  quarre';  be- 
cause the  second  power  of  a  factor  represents  the  surface  of  a  square, 
whese  side  is  represented  by  the  factor  (Geom.  §§124,  171,  177). 
(c)  Gr.  Kii/3oc;  because  the  third  power  of  a  factor  represents  the  solid 
content  of  a  cube,  whose  edge  is  represented  by  the  factor. 


INTRODUCTION.  [§  23; 

that  they  have  such  exponents.  So  a  power,  whose  expo- 
nent is  an  even  number,  is  frequently  called  an  even  pow- 
er ;  one  whose  exponent  is  an  odd  number,  an  odd  power. 

§  23.  One  of  the  equal  factors  (§12)  of  a  quantity 
is  called  its  root. 

a.)  A  root  is  called  the  second  or  square,  the  third  or 
cube,  the  fourth,  the  nth,  according  as  it  is  one  of  two,  three, 
four,  or  n  equal  factors,  which  produce  the  given  quantity ; 
i.  e.  into  which  the  given  quantity  is  separated. 

Thus  2  is  the  third  or  cube  root  of  8,  because  it  is  one  of 
three  equal  factors,  which  produce  8.     So  a  is  the  third 

root  of  a3,  the  fourth  root  of  a4  ;  a2  is  the  second  or  square 
root  of  a3,  because  it  is  one  of  the  two  equal  factors,  into 
which  a3  may  be  separated  (§  12.  c). 

b.)  A  root  of  any  quantity  is  properly  expressed  by  writ- 
ing the  quantity  under  a  fractional  exponent,  whose  numer- 
ator is  unity,  and  whose  denominator  is  equal  to  the  num- 
ber of  the  root  (§  12.  c).  For  this  denotes,  that  the  quan- 
tity under  the  fractional  exponent  is  separated  into  so  many 
equal  factors,  as  there  are  units  in   the  denominator,  ami 

that  only  one  of  them  is  taken.     Thus, 

x  i 

The  second  or  square  root  of  a  is  cr  ;  that  of  a2  is  («2) 

—  a      -  (§  20)  —  a ;  the  third  or  cube  root  of  a  is  a6 ;  that 

of  a2  is  02)*=cA 

c.)  The  principle  of  §  12.  b,  c  may,  therefore,  be  express- 
ed as  follows : 

A  fractional  exponent  shows,  either  that  the  root  of  the 
base,  denoted  by  the  denominator,  is  raised  to  the  power  de- 
noted by  the  numerator  (§  12.  b);  or,  that  the  base  being 
raised  to  the  power  denoted  by  the  numerator,  the  root  de- 

5 

noted  by  the  denominator  is  taken  (§  12.  c).     Thus,  a2  ex- 
presses the  fifth  power  of  the  second  or  square  root  of  a ; 

or  the  square  root  of  the  fifth  power  of  a ;   so  8^  is  equal 


§  23.]  FACTORS  AND  POWERS.  37 

to  the  square  of  the  cube  root  of  8 ;   or  to  the  cube  root  of 
the  square  of  8. 

d.)  A  root  is  also  frequently  indicated  by  the  radical* 
sign,  */c,  placed  before  the  quantity,  with  a  number  over 
the  sign,  to  show  the  number  of  the  root.  In  expressing 
the  second  or  square  root,  however,  the  number  is  more 
frequently  omitted ;  and,  accordingly,  wherever  the  sign 
stands  without  a  number  over  it,  it  must  always  be  under- 
stood to  denote  the  square  root.  Thus, 
x 

.y  4  =  42  =  the  second  or  square  root  of  4. 

V8  =  8*  =  "  third  or  cube  «     «  g. 

«V«  ==■«*  =  "  fifth  "     "  a. 

Note.  Either  of  these  forms  of  expressing  the  root,  may  be  used 
at  pleasure,  and  both  should  be  made  familiar.  The  fractional  ex- 
ponent is,  however,  generally,  more  convenient  than  the  radical  sign; 
and  is,  besides,  to  be  preferred  because  it  exhibits  roots  as  a  class  of 
poivers,  and  enables  us  to  refer  the  operations  upon  roots  to  tho  gen- 
eral principles,  which  govern  the  operations  upon  powers.  Quanti- 
ties written  under  a  radical  sign  are  frequently  called  radical  quan- 
tities. 

e.)  As  the  product  of  an  odd  number  of  positive  factors 
is  positive,  and  of  negative  factors,  negative  (§11.  Note  2)  ; 
hence,  an  odd  root  (i.  e.  a  root  denoted  by  an  odd  number) 
of  any  quantity  must  have  the  same  sign  as  the  quantity  it- 
self.    Thus, 

(+a) 3  — +«3>  and  ( — a)3  = — a3. 
(+a3) *  or  V+a3  =  +a  ; 

JL 

and  (— a3)  3,  or  3y— a3  =  —  a. 

/.)  Again,  since  the  product  of  an  even  number  either  of 
positive  or  of  negative  factors  is  always  positive  (§11.  Note 
2) ;  therefore, 

(1.)  Every  even  root  (i.  e.  every  root  denoted  by  an 

(d)  Lat.  radix,  root,  (e)  A  modified  form  of  the  letter  r,  the 
initial  of  radix. 

ALG.  4 


38  INTRODUCTION.  [§  24. 

even  number)  of  a  positive  quantity  may  be  either  positive 
or  negative. 

This  character  of  the  root  is  denoted  by  the  double  sign 
±  (read  plus  or  minus).     Thus, 

(+a)(+a)=:-(-a2,  and  (—«)(—«)  =  -(-a2. 

(a2)2  or  ya2  ==  ±a. 
(2.)  An  evera  root  of  a  negative  quantity,  can  be  neither 
positive  nor  negative,  and  therefore  does  not  really  exist, 
and  is  said  to  be  imaginary.     For  neither  (-\-a)(-\-a),nor 
( — a)( — a)  can  produce  — a2. 

§  24.  It  is  evident,  from  the  definition  of  a  power,  that 
whatever  has  been  demonstrated  of  quantities  with  expo- 
nents is  true  of  powers.  Hence  we  have  the  following 
rules. 

RULE  I. 

a.)  To  multiply  powers  of  the  same  quantity  to- 
gether. 

Add  their  exponents.     §  15.  Cor.  II. 

a4.a3  =  what?     a~\aG  ?     Ac2  ?     aAaf*?     3*.3~3  ? 

RULE  II. 

b.)  To  divide  a  power  of  a  quantity,  by  any  pow- 
er of  the  same  quantity. 

Subtract  the  exponent  of  the  divisor  from  that  of 
the  dividend.     §  16.  Cor.  III. 

_=what?     —  ?     ^?     gj?     -t?     ~? 

RULE  III. 

c.)  To  find  the  reciprocal  of  a  power. 
Change  the  sign  of  the  exponent.     §  18. 


§  25,  26.]  FACTORS  AND  POWERS.  39 

What  is  the  reciprocal  of  a  ?     of  a4  ?     of  10  ?     of  10 2  ? 
of  lO-i?     ofi^?     ofa2ar2?     ofx^?     ofa^cc-5? 

RULE  IV. 

d.)  To  find  any  power  of  a  power. 
Multiply  the  exponent  of  the  given  power  by  that  of 
the  required  power.     §  20. 


3 


•- 


1.  What  is  the  second  power  of  a2  ?     of  a1  ?'    of  162  ? 

of  16"^?     of— a? 

2.  (a2)-2  —  what?     («*)e?     (— 10)3  ?     (or*)*? 

3.  (a*)*=what?  (10g)2?  (R2)~h  (xrf?     (lO^h 

§  25.  e.)  The  last  rule  obviously  applies  equally  to  the 
finding  of  a  root ;  i.  e.  a  power,  whose  exponent  is  unity  di- 
vided by  the  number  of  the  root  (§  23.  b).  But  to  multiply 
by  such  a  fraction  is  the  same  as  to  divide  by  its  denomina- 
tor. Hence  we  have  the  common  rule  for  finding  a  root  of 
a  power : 

Divide  the  exponent  of  the  power  by  the  number  of  the 
root. 

What  is  the  third  root  of  a3  ?     of  a2  ?     of  a  ?     of  106  ? 
What  is  the  second  root  of  10 4  ?     of  x3  ?    of  x6  ?    of  2  ? 

What  is  the  third  root  of  10  2?    ofa^?    ofa^?    of  af"3? 

Note.  It  should  be  borne  in  mind,  that  the  word  power  is  used, 
!n  all  these  cases,  in  the  ividest  sense;  and  that  the  rules  are  equally 
applicable  to  all  the  classes  of  powers  specified  in  §  22. 

§  26.  A  quantity,  whose  value  is  determined  by  the 
value  assigned  to  another  quantity,  is  said  to  be  a 
function7  of  that  other  quantity. 

Thus,  a2,  a3,  a*,  are  functions  of  a,  because  their  value 
depends  upon,  and  is  determined  by,  the  value  assigned  to 

(/)  Lat.  functio,  from  fungor,  to  perform,  as  depending  on  the 
performance  of  certain  operations  upon  another  quantity. 


40  INTRODUCTION.  [§  27,  28. 

a.     Thus,  let  a  =  1,  then  a2  =  l;   if  a  =  2,  then  a-  =  4; 
if  a  =  10,  then  a2  =  100. 

So,  if  2«  =  a;2,  or  u  —  2x,  or  u  —  3x,  then  m  is  a  function 
of  a; ;  or,  as  it  is  usually  expressed,  u  =  F(x),  or  u  =zf(x)  ; 
where  F  and  f  are  not  factors,  but  mere  abbreviations 
for  the  words  function  of 

A  power  is  a  function  of  a  quantity,  expressed  by 
an  exponent  written  over  the  quantity ;  i.  e.  an  expo- 
nential function  of  the  quantity. 

§  27.  A  power  is  said  to  be  of  such  a  degree^  as 
is  indicated  by  the  exponent.     Thus, 

a3  is  of  the  third  degree;    «2,  of  the  second;    a  of  the 

JL 

first;  «~4,  of  the  minus  fourth;  and  a2,  of  the  one-half  de- 
gree. 

§  28.  The  degree  of  a  term  is  equal  to  the  sum  of 
the  exponents  of  its  literal  factors.     Thus, 

a,  x,  2x,  3a2:*:-1,  a3b°x~2  are  of  the  first  degree. 

L    I        3      _X.       2.    1 

So  a-x-,  a-.x  -,  a'sx's  are  of  the  first  degree. 

\      3 

2ax,  2px,  y2,  a*b~2,  p2x2  are  of  the  second  degree. 
3a-x  is  of  the  third,  and  Aa3x,  of  the  fourth  degree. 

i  a 

a2  is  of  the  one  half,  and  a3,  of  the  two  thirds  degree. 

a~2x~2,  and  a3x~7  are  of  the  minus  fourth  degree. 

1.  9c<564c-3  is  of  what  degree?     15x2y2  ?    5a3&X&cy? 

5     3  2      7  5     _2  1     _2_ 

«-&%?     3a%3?     a3 x  3?     a3«3?     or3*0  ? 

Note.    A  term  ia  also  sometimes  said  to  have  as  many  dimen- 
sions71 as  there  are  units  in  its  degree. 


(g)  Fr.  degre',  from  Lat.  gradus,  step,  (h)  Lat.  dimensio,  from 
dimetior,  io  measure.  The  use  of  this  word  resulted  from  taking  a 
factor  to  represent  a  line,  and,  consequently,  a  product  of  two  fac- 
tors to  represent  a  surfaco,  and  one  of  three  factors,  to  represent  a 
solid.  The  factors  were  therefore  regarded  as  the  dimensions,  or 
measures  of  the  magnitudes.  See  Geom.  §§3,  170,177.  The  word 
is,  of  course,  not  strictly  applicable  to  any  term  of  a  degree  higher 
than  the  third  (Geom.  §  2.  a),  or  lower  than  the  first. 


§  29,  30.]  FACTORS  AND  POWERS.  41 

a,)  In  estimating  the  degree  of  a  fractional  term,  the 
exponents  of  the  letters  in  the  denominator  must,  of  course, 
be  regarded  as  negative  (§  14, 16),  and  subtracted  from  the 
sum  of  the  exponents  of  the  letters  in  the  numerator.    Thus, 

3  n  -  h  *  ft  I)  y 

—  is  of  the  first  degree  ;  — -r-  and ,  are  of  the  second. 

a2  a  c 

b.)  A  term  is  said  to  be  of  the  first,  second,  third  or  nth 
degree  with  respect  to  a  particular  letter  or  letters,  when  it 
contains  the  first,  second,  third,  ^or  nth.  degree  of  the  letter 
or  letters.     Thus, 

3a2x,  and  a~1x  are  of  the  first  degree  with  respect  to  x. 

b2x2  and  ax2  are  of  the  second      "  "  x. 

x 
abx2  and  +/x  are  of  the  one  half  "  "  x. 

a2x°,  and  abc  are  of  the  zero         "  "  x. 

axy  is  of  the  first  degree  with  respect  to  either  a,  x  or 
y  ;  and  of  the  second  degree  with  respect  to  x  and  y,  or 
any  two  of  the  letters  ;  while  it  is  of  the  third  degree  with 
respect  to  all  the  letters. 

§  29.  Terms  of  the  same  degree  are  said  to  be  ho- 
mogeneous'. 

Thus,^,  2x,  and  \a2x~x  are  homogeneous.     So  a3,  Sax-. 

.    ...  ab  B2x" 

xyz ;  m  like  manner,  y,  — ,  and    .2  f/x, 

1.  Are  A2y2  and  B2x2  homogeneous?  x3,  2y2  and  x'r 
JR2  and  sin  a  sin  b? 

§  30.  Terms,  which  consist  of  the  same  literal  fac- 
tors, with  the  same  exponents  (i.  e.  each  letter  being 
of  the  same  degree  in  the  several  terms),  are  called 
similar  or  like  terms. 

Thus,  2xy,  8xy,  and  3yx  are  similar  terms  ;  so  3x  2y,  and 
\x2y.  But  3x2y  and  3xy2  are  not  similar,  because,  though 
the  letters  are  the  same,  they  have  different  exponents  in 
the  two  terms.     Are  3x2y  and  3xy2  homogeneous  ? 

(i)  Gr.  buoyevris,  compounded  o( 6/jor,  like,  and  yevoc,  kind, 

*1 


J 


42  INTRODUCTION.  [§  31-33. 

Are  a2b2  and  x2y2  similar?     Are  they  homogeneous? 

a.)  Thus  terms  may  be  homogeneous  without  being  sim- 
lar,  but  they  cannot  be  similar  without  being  homogeneous. 

b.)  Terms,  in  which  the  same  letter,  with  the  same  expo- 
nent, enters,  are  sometimes  said  to  be  similar  with  respect 
to  that  letter.  Thus  the  terms  ax,  obex  and  c*x  are  similar 
with  respect  to  x. 

MONOMIALS  AND  POLYNOMIALS. 

§  31.  A  quantity  consisting  of  one  term,  is  called  a 
monomial*;  of  more  than  one,  a  polynomial'.  A 
polynomial  of  two  terms  is  called  a  binomial'";  one 
of  three  terms,  a  trinomial". 

Thus,  2ax,  a,  a2b2,  abc  are  monomials  ;  so  aby.xy-^z  ; 
a-\-b,  a — b,  x- — y2  are  binomials;  a-\-b-\-c,  a2±2ax-\-x2 
are  trinomials. 

§  32.  A  polynomial  is  said  to  be  homogeneous. 
when  all  its  terms  are  homogeneous  (§  29.). 

Thus,  a3±3a2b-\-3ab2±b3,  A2y2-{-B2x2—A2B2  are  ho- 
mogeneous polynomials. 

1.  Is  x2-\-y2 — R2  homogeneous?  x5±ox4y-\-lQx:iy- 
±\0x2yZ-\-oxy±±y5  ? 

b3  b3 

2.  Is  a2A-b3  homogeneous?     a2  1       ?     a3-\ ? 

c  '   a  a 

§33.  When  the  several  terms  of  a  polynomial  contain 
different  powers  of  any  letter  or  letters,  it  is  generally  con- 
venient to  arrange  the  terms  according  to  the  powers  of 
some  one  letter ;  that  is,  to  write  the  term  containing  either 
the  highest,  or  the  lowest  power  of  the  letter  first,  and  the 
other  terms  successively,  according  to  the  order  of  their  ex- 

(fc)  monome,  from  Gr.  fiovoc,  alone,  and  bvojia,  Lat.  nomen, 
name;  as  being  expressed  by  a  single  name  or  term.  (I)  poly- 
nome  from  Gr.  -oA<'< ,  many,  and  oroiia,  name,  (m)  Lat.  bis,  twice, 
and  nomen,  name,     (n)  Lat.  tres,  three,  and  nomen,  name. 


§  34.1  REDUCTION  OF  POLYNOMIALS.  43 

ponents  ;  from  highest  to  lowest,  or  from  lowest  to  highest. 
If  the  highest  exponent  is  placed  first,  the  terms  are  said  to 
lie  arranged  in  a  descending  series,  or  according  to  the  des- 
cending powers  of  the  letter ;  if  the  lowest  is  placed  first. 
the  arrangement  is  said  to  be  in  an  ascending  series,  or  ac- 
cording to  the  ascending  powers  of  the  letter. 

Thus,  a2-\-2ab-\-b~  is  arranged  according  to  the  ascend- 
ing powers  of  b,  and  according  to  the  descending  powers  of 
a. 

1.  Arrange  oaQb-\-3ab2-\-b3-\-a3  according  to  the  des- 
cending powers  of  a ;  of  b. 

2.  Arrange  rqxn~2Jrxn-\-px"-1-\-rxn-3  according  to  the 
descending  powers  of  a?. 

Note.  The  letter,  according  to  whose  powers  the  terms  of  a  po- 
lynomial are  arranged,  is  frequently  called  the  letter  of  arrange- 
ment. When  there  is  no  special  reason  for  a  different  order,  it  is 
generally  convenient  to  write  the  letters  of  each  term  in  the  order  of 
the  alphabet;  and  also  to  take  the  first  of  those  letters,  as  the  letter 
of  arrangement. 

REDUCTION  OF  POLYNOMIALS. 

§  34.  A  polynomial,  which  contains  similar  terms. 
can  be  reduced  to  a  simpler  form. 

This  is  done  according  to  the  principles  of  §  4.  Thus, 
in  the  polynomial  4a — 6a-f-9a — 3a,  4a  and  9a  are  to  be 
added,  and  6a  and  3a  are  to  be  subtracted.  It  is  usually- 
most  convenient  to  brino;  together  the  terms  which  are  to 
be  added,  and  also  the  terms  which  are  to  be  subtracted, 
and  then  take  the  less  from  the  greater.  If  the  quantity 
to  be  added  is  greater  than  that  to  be  subtracted,  the  result 
is  to  be  added  ;  i.  e.  is  positive.  If  the  quantity  to  be  sub- 
tracted is  greater  than  that  to  be  added,  the  result  is  to  be 
subtracted ;  i.  e.  is  negative  (§  4.  a,  b).  Hence,  for  reduc- 
ing or  simplifying  a  polynomial  containing  similar  terms, 
we  have  the  following 


44  INTRODUCTION-  [§  3-i. 

RULE. 

Add  together  the  coefficients  of  such  similar  terms 
as  have  the  sign  +;  and  then  the  coefficients  of  such 
as  have  the  sign  — ;  take  the  less  of  these  sums  from 
the  greater ;  and  prefix  the  remainder,  with  the  sign  of 
the  greater,  to  the  common  letter  or  letters.     Thus, 

4a — 6a-\-9a — 3a  —  4a-f-9a — 6a — oa  =  13a — 9a  =  4a. 

a.)  Terms  of  a  polynomial,  which  are  not  similar,  will, 
of  course,  remain  as  they  were ;  each  being  preceded  by  its 
own  sign. 

Reduce  the  following  polynomials  to  their  simplest  form. 

1.  a2—ab—ab-\-b2.  Ans.  aj—2ab+b2. 

2.  a2-\-ab—ab—b2. 

b.)  There  may  be  several  sets  of  similar  terms  in  the 
same  polynomial.  In  that  ease,  the  above  method  must,  of 
course,  be  applied  to  each  set  separately.     Reduce, 

1.  5a_|_G6— 7x— 8J-f3a— 4a+2ic+9a— 3x. 

2.  «*_ 3a3x-{-3a2x2— ax3— a3x-\-3a2x2— 3ax3-{-x*. 

3.  l-\-x—l-\-x.  4.   1-4-x+l— x. 

5 .  y  2  -\-x  2—px-\-\p  2  — x  -  —px — \p  2 . 

6.  2bx-\-2x2— b2—  2bx— x2. 

7.  a3+a2b+ab2—a2b—ab2—!>3. 

c.)  If  a  polynomial  contains  several  terms  similar  in  res- 
pecl  to  a  certain  letter  (§  30.  b),  the  same  principle  will  ob- 
viously apply.  Thus,  the  terms  ax-\-bx — 2cx,  are  similar 
in  respect  to  x.  Now,  a  times  x,  2)lus  b  times  x,  minus  2c 
times  x  is  evidently  the  same  as  x  taken  a-\-b — 2c  times, 
which  (§  2.  h)  is  expressed  (a-\-b—2c)x.  Hence,  we  may 
write  the  coefficients,  whether  numerical  or  literal  (§  9.  b), 
of  the  common  letter  or  letters  in  the  several  terms,  in  or- 
der, with  the  signs  of  the  terms ;  enclose  the  whole  expres- 
sion, so  formed,  in  a  parenthesis,  or  put  it  under  a  vinculum  ; 


}  35-37.]  EQUATIONS.  45 

and  write  the  common  letter  or  letters,  without  the  paren- 
thesis or  vinculum,  as  a  separate  factor.     Eeduce, 

1.  A2x2— c2x2.  Ans.  (A2— c2)x2. 

2.  2px"-\-px — px". 

3.  j^y z+B2x2—A 2 B2—A  V 2— B-x" 2-\-A2B2. 

§  3-5.  The  numerical  value  of  an  algebraic  expres- 
sion is  the  result  obtained  by  assigning  particular 
values  to  the  letters,  and  performing  the  operations 
indicated  by  the  symbols.     Thus, 

Let  a  =  10,  and  b  =  5,  then  a-f-5  =  15;  (a-\-b)2=z 
152  =  225  ;  a2-\-2ab+b2  =  102-f  2.10.5+52  —  225. 

1.  Let«  =  10  and  5  =  4,  and  find  the  value  of  a5-\- 
3a2b+3ab?-{-b3  ;  (a—b)2;  a2—b2;  (a+b)(a—b). 

2.  Find  the  value  of  the  same  expressions,  when  a  =  8, 
and  b  =  3;  when  a  =  20,  and  5  =  5;  when  a  =  10,  and 
5=10;  when  a  =  10,  and  5  =  9  •  when  a  =  1,  and  5  =  1. 

3.  Find  the  value  of  y^-2x—  4,  when  ?/  =  10,  and  #  =  3  ; 
when  y  =  8,  and  :c  =  2 ;  when  #  =  4,  and  x  =  0. 

EQUATIONS. 

§3(5.  The  expression  of  equality  between  two 
quantities  constitutes  an  equation"  ;  as, 

5+4  =  10—1 ;  dmXa"  =  am+n ;  3x  =  15  ;  ax  =  b. 

a.)  The  two  quantities  themselves  are  called  the  mem- 
bers1' or  sides  of  the  equation.  The  member  on  the  left 
of  the  sign  is  styled  the  Jirst,  and  that  on  the  right,  the  sec- 
ond member. 

b.)  Most  of  the  investigations  and  reasonings  of  Algebra 
are  carried  on  by  means  of  equations. 

§  37.  c.)  The  simplest  form  of  equation  is  that,  in  which 

(o)  Lat.  sequatio,  from  icquo,  to  make  equal,     (p)  L.  membruin. 
limb. 


'16  INTRODUCTION.  [§  38,  39. 

the  two  sides  are  precisely  alike ;  as  10  =  10 ;  x-\-2  ±=  x~\-2. 
These  are  called  identical'1  equations. 

d.)  Another  class  of  equations  we  have  already  employ- 
ed, in  expressing  the  results  of  operations,  or  the  truth  con- 
tained in  such  results.  Thus,  a3X«4  =  «;;  aman  =  a'"+" ; 
l-7-x=. x~l  (§17).  These  may  be  called  absolute  equa- 
tions ;  inasmuch  as  their  truth  has  no  dependence  upon  the 
value  assigned  to  a,  x,  m  or  n.  The  second  member 
necessarily  results  from  the  operation  indicated  in  the  first. 

§  38.  e.)  In  another  class  of  equations,  there  is  no  abso- 
lute and  essential  equality  between  the  members ;  but  they 
are  equal  only  on  the  condition,  that  some  particular  value 
or  values  be  given  to  one  or  more  of  the  quantities  involv- 
ed. Equations  of  this  kind  may  be  called  conditional  equa- 
tions. Thus,  2x  =10  is  a  conditional  equation,  in  which 
the  equality  of  the  members  depends  on  the  condition,  that 
x  shall  be  equal  to  5.  If  4  were  taken  as  the  value  of  x 
the  two  members  would  not  be  equal ;  we  should  have  8 
on  one  side,  and  10  on  the  other.  But  taking  x=^5,  then 
2X5  =  10,  or  10  =  10. 

f.)  A  conditional  equation,  moreover,  itself  furnishes  the 
means  of  investigating  and  ascertaining  the  value  which 
must  be  given  to  x,  in  order  that  the  members  may  be 
equal ;  that  the  equation  may  become  absolute  or  identical. 
For  lx  is  obviously  half  as  much  a3  2x ;  if  then,  we  have 

2x=zl0, 
we  shall  have  a:  =  ^  of  10  =  5,  the  necessary  value 

of  x,  as  above. 

Conditional  equations  may  therefore  be  called  equations 
of  investigation. 

§  39.  Any  quantity,  to  which  a  particular  value  must  be 
given,  in  order  to  render  the  members  equal,  is  called  an 
unknown  quantity  (§  1.  c,  N).  That  value  of  an  unknown 
quantity,  which  renders  the  members  equal,  is  called  a  root 

(q)   Fr.  identique,  from  Lat.  idem,  the  same. 


§40,41.]  EQUATIONS.  47 

of  the  equation.  "When  this  value  is  substituted  for  the  un- 
known quantity,  it  is  said  to  satisfy  or  verify  the  equation. 
The  process  of  finding  the  root  of  an  equation  is  called  solv- 
ing the  equation. 

Note.  When  equations,  without  any  specification,  are  spoken  of, 
or  when  the  subject  of  equations  is  spoken  of,  as  a  branch  of  alge- 
braic science,  the  expression  must,  in  general,  be  understood  to  im- 
ply conditional  equations. 

§  40.  Conditional  equations  are  distinguished  into 
orders,  according  to  their  degree. 

a.)  The  degree  of  an  equation  depends  on  the  degrees  of 
its  terms  with  respect  to  the  unknown  quantity  or  quanti- 
ties (§  28.  b)  ;  and  is  determined  by  the  range  of  those  de- 
grees from  lotvest  to  highest. 

b.)  The  full  consideration  of  this  subject  would  involve 
the  consideration  of  equations  containing  negative  and  frac- 
tional powers  of  the  unknown  quantities. 

c.)  For  the  present,  however,  it  is  sufficient  to  consider 
those  equal  ions  only,  in  which  the  exponents  of  the  un- 
known quantity  or  quantities  are  all  integral,  and  in  which 
the  least  of  those  exponents  is  zero. 

d.)  In  this,  case,  the  degree  of  the  equation  is  ihe 
same  as  the  highest  degree  of  its  unknown  quantity 
or  quantities.     Thus, 

ax  ~b,  2x—  10,  and  x-\-y  —  10  are  of  the  first  degree. 

ax2  =b,  x'2-\-3x  =  10,  and  xy  =  20  are  of  the  second 
degree. 

§  41.  We  shall,  at  present,  confine  ourselves  to  the  con- 
sideration of  equations  containing  but  one  unknown  quanti- 
ty;  subject  also  to  the  limitation  mentioned  above  (§  40.  c). 

These  equations  are  said  to  be  of  the  same  degree 
as  the  highest  power  of  the  unknown  quantity  which 
they  contain.     Thus, 

3x=  18 ;  ax  —  b  are  equations  of  the  first  degree. 


I  INTRODUCTION.  [§  42,  43. 

x-  —  9  ;  ax2-\-bx  =  c  are  equations  of  the  second  degree. 

ax*-\rbx*-j-cxz=hi  x3=S      «  third  " 

a^+6x3=rc;  x4  =  16  «  fourth        « 

xn+^"-1+&c.  =  7«  is  "  nth  « 

Note.  Equations  of  the_/irs£  degree  are  sometimes  called  simple 
equations ;  those  of  the  second  degree,  quadratic*;  those  of  the  third, 
cubic  ;  and  those  of  the  fourth,  biquadratic*. 

\  42.  All  reasoning  by  means  of  equations  pro- 
ceeds upon  a  single  axiom",  or  self-evident  truth  ; 
viz.  Equal  quantities,  equally  affected,  remain 
equal.     Geom.  20. 

The  meaning  of  this  axiom,  which,  though  not  always 
expressed  in  words,  is  assumed  in  all  mathematical  opera- 
tion, may  be  illustrated  by  a  few  familiar  examples.  Thus, 
3x5  =  15  is  an  equation.  Adding  2  to  both  sides,  we 
have  3X5+2  =  lo-f-2.  Subtracting  4  from  both  sides  of 
the  first  equation,  we  have  3x5 — 4=15 — 4.  In  like 
manner,  we  might  multiply  or  divide  both  sides  by  the 
same  quantity,  and  obtain  equal  products  or  quotients. 

Hence,  if  both  members  of  an  equation  be 

a.  increased  by  the  addition  of, 

h.  diminished  by  the  subtraction  of,  \  equal  quantities,  the  re- 

c.  multiplied  by,  [      suits  will  be  equal. 

d.  divided  by,  J 

§  43.  I.)  1.  Given  x— 3  =  7  ;  to  find  the  value  of  x. 

Add  3  to  each  side  ; 
then  a;— 3+3  =  7+3.  §  42.  a. 

or  x  =  10,  the  value  required. 

2.  Given  x — 5  =  4,  to  find  the  value  of  x. 

Ans.  x  =  9. 

3.  Given  x— 1G  =  20,  to  find  the  value  of  x. 


(s)  Lat.  quadra,  square,  (t)  Lat.  bis,  tivice,and  quadra,  square. 
(»)  Gr.  utjiu/ia,  from  dftou,  to  deem  worthy,  suppose,  take  for  grant- 
ed. 


1 44.]  EQUATIONS.  49 

II.)  4.  Given  x-{-3  =  7,  to  find  x. 

Subtract  3  from  each  side ; 
then  x+3— 3  =  7—3.  §  42.  b. 

or  x  =  4,  the  value  required. 

5.  Given  x-|-10  =  15,  to  find  a\  ^4ns.  x=  5. 

6.  Given  2a;  =  10-f-#,  to  find  x. 
Subtract  x  from  each  side ; 

then  2x— x  =  10-\-x— x.  §  42.  b. 

or  a?  =  10,  the  value  required. 

7.  Given  3x — 10  =  10-(-2x,  to  find  the  value  of  x. 

Note.  To  verify  or  prove  these  results,  we  have  only  to  intro- 
duce, into  the  given  equation,  the  value  found  for  the  unknown  quan- 
tity in  place  of  the  unknown  quantity  itself.  Thus,  in  example  1 
above,  substituting  for  x  its  value  found,  we  have 

10 — 3  =  7,  an  absolute  equation.     See  §39. 

Verify  the  otber  equations  in  like  manner. 

§  44.  Thus  we  see  that  the  application  of  §  42.  a  and  b 
causes  any  term,  which  stands  on  one  side  of  an  equation, 
preceded  by  the  sign  either  of  addition  or  subtraction,  to 
disappear  from  that  side,  and  to  reappear  on  the  other 
side  with  the  opposite  sign.  Thus,  in  §  43.  1,  by  adding  3 
to  both  sides,  and  reducing,  —3  is  canceled  in  the  first 
member,  and  -f-3  appears  in  the  second ;  so,  in  §  43.  4,  -f-3 
is  canceled  in  the  first  member,  and  —3  appears  in  the 
second. 

This  is  called  transposition1'.  For  the  same  effect  would 
obviously  have  been  produced,  if  we  had  simply  removed 
the  term  from  the  one  side,  and  written  it  with  the  oppo- 
site sign  upon  the  other.  In  fact,  removing  — 3  (i.  e.  ceas- 
ing to  subtract  3)  from  the  first  member  (§  43.  1)  increases 
that  member  by  3  ;  3  must,  therefore,  be  added  to  the  sec- 
ond member,  to  preserve  the  equality.  So  (§  43.  4),  re- 
moving -f-3  (i.  e.  ceasing  to  add  3)  diminishes  the  first 

(t>)  Lat.  transpositio,  from  transpono,  to  place  beyond,  carry  ovi  r, 
ALG.  5 


i ) 


INTRODUCTION.  [§  45. 


member  by  3 ;   3  must,  therefore,  be  subtracted  from  the 
second  member.     Hence, 

Any  quantity  may  be  transposed  from  one  side  of  an 
equation  to  the  other,  if,  at  the  same  time,  we  change 
sign. 

If  we  transpose  all  the  terms  of  an  equation,  the  signs 
will  all  be  changed,  and  the  members  will  still  be  equal. 
Hence, 

Corollary.  The  signs  of  all  the  terms  of  an  equation 
may  be  changed  at  pleasure,  without  affecting  the  equality 
of  the  members. 

It  is  also  evident  from  §  42.  a,  b,  that  the  same  quan- 
tify, with  the  same  sign,  occurring  on  both  sides  of  an  equa- 
•.  may  be  suppressed. 

c.)  The  object  of  transposition  is,  in  general,  to  bring  all 
the  terms  containing  the  unknown  quantity  to  stand  on  one 
^ide  of  the  equation  ;  and  all  the  known  terms,  upon  the 
i  ither.  The  polynomials  so  formed  should,  of  course,  be 
reduced  to  their  simplest  form  (§  34). 

1.  Given  8a+4  =  72+12,  to  find  the  value  of  x. 

2.  Given  loy — 3  =  12+5?/ — 3+9?/,  to  L find  the.  value 
of  y.  Ans.  y=  12. 

3.  Given  2x-{-a-\-b  =  ox-\-2a — 2x,  to  find  x. 

Ans.  x  =  a — b. 

4.  Given  4a+-3a+25  =  4a+3a+S,  to  find  x. 

5.  Given  2>x— 10  =  5+2x— 15,  to  find  x.  Ans.  x  =  0. 

6.  Given  2x — 10  =  x—  15,  to  find  x.      Ans.  x  =  —  5. 

x 

^  45.  1.   Given  -+3  =  8,  to  find  x. 
4 

Transpose  3 ; 

then  i  —  5-  §44> 

4 

Multipl}'  by  4 ; 
then  x  =  20.  §42.  c. 


§  46.]  EQUATIONS.  51 

To  verify  this  equation,  substitute  20  for  x,  and  we  hav< 

20 

_-J-3  =  8;  or  5+3  =  8,  an  absolute  equation. 
4 

2.  Given  ^—5  =  3,  to  find  x.  Ans.  x  =  24. 

o 

X 

3.  Given  -  —  ^x—  2  =  0,  to  find  x. 

Multiply  by  2  ; 
then  x—  %x— 4  =  0.  -'.  c. 

Multiply  by  3  ; 
then  3a;— 2x— 12  =  0. 

Reducing,  a;— 12  =  0.  §  34. 

x  =  12.  H4- 

4.  Given  -  —  -  =  5,  to  find  x. 

6      4 

§  46.  Thus,  if  a  quantity  in  an  equation  be  divided  by 
any  number,  the  application  of  §  42.  c  enables  us  to  free  it 
from  its  divisor,  i.  e.  to  clear  the  equation  of  fractions. 

The  terms  of  an  equation  may,  therefore,  be  freed 
from  divisors,  or,  in  other  words,  an  equation  may  be 
cleared  of  fractions,  by  multiplying-  all  the  terms  of 
the  equation  by  the  denominators  of  the  fractional 
terms. 

Note.  The  equation  is  to  be  multiplied  first  by  one  of  the  denom- 
inators, and  then  the  resulting  equation  by  another,  and  so  on,  till  all 
the  terms  containing  the  unknown  quantity  become  whole  numbers 
In  this  process  improper  fractions  may  always  be  reduced  to  whole 
numbers,  whenever  it  can  be  done;  and  no  more  multiplication.- 
should  be  performed,  than  are  necessary  to  clear  the  equation  of 
fractions. 

a.)  The  same  effect  would  obviously  be  produced  by  mul- 
tiplying all  the  terms  of  the  equation,  by  any  common  mul 
tiple  of  the  denominators ;   i.  e.  by  any  number  which  th< 
denominators  will  all  divide  without  a  remainder.     For  ii 
^nominator  will  divide  the  multiplier,  it  will  necessarih 


52 


INTRODUCTION.  [§  47. 

divide  the  product  of  its  own  numerator  into  that  multipli- 
er.    Thus, 


Let                   -  +  -4-^  — £^_i_i 

Multiply  by  30 ; 

en                     1  ox+Gx+ox  =  25a;+30, 

§42.  e 

X=r30. 

§§  44,  34 

Given  --j--  +  -__-__f-i,  to  find  x. 

Note.  A  common  multiple  may  easily  be  found  by  trial.  Thus, 
in  the  above  equation,  try  8,  the  largest  denominator,  and  see  if  the 
other  denominators  will  divide  it  without  a  remainder.  We  find,  that 
3  and  6  will  not  so  divide  it.  Then  multiply  it  by  2;  still  we  do  not 
obtain  a  multiple  of  those  denominators.  Multiply  8  by  3,  and  all 
the  denominators  will  divide  the  product;  24,  therefore,  is  a  multi- 
plier, which  will  clear  the  equation  of  fractions.  It  is  important  to 
employ  the  smallest  multiplier,  which  will  accomplish  the  object. 

b.)  By  clearing  of  fractions,  the  coefficients  of  the  un- 
known quantity  all  become  integral ;  and  the  polynomial, 
formed  by  collecting  all  the  terms  containing  the'  unknown 
quantity  into  one  member,  is  the  more  easily  reduced  to  a 
simpler  form  (§34). 

Note.  Whether  transposition  or  clearing  of  fractions  be  first  per- 
formed, is  indifferent.  Any  course  may  be  taken  in  this  respect 
which  is  found  convenient.  See  §45.  1,  3.  The  whole  process  of 
clearing  effractions,  transposition,  and  reducing  the  polynomial  mem- 
bers to  their  simplest  form,  is  sometimes  called  the  reduction  of  tin 
equation. 

1.  Given  —  -f-  —  =  —  -)-5,  to  find  x.     Ans.x  —  ^- 

3         4        3 

2.  Given  -  +  -  =  —  +4,  to  find  x. 

Zoo 

3.  Given  x-f-10  —  —  +  ^  +50,  to  find  x- 

§  47.  1.  Given  2x— 7  =  9— Gar,  to  find  x. 
Transpose ; 
then  8.*=  16.  §44. 


§  48.]  EQUATIONS.  53 

Divide  the  terms  by  8  ; 
then  x  =  2.  §  42.  d. 

2.  Given  3x-\-5  =  x-\-20,  to  find  x.  Ans.  x  —  l\. 

3.  Given  4x — 8  —  40 — 2x,  to  find  x. 

Thus  it  is  obvious,  that,  when,  by  any  means,  a 
single  term  containing  the  unknown  quantity  is 
made  to  constitute  one  member  of  an  equation,  while 
the  other  member  consists  wholly  of  known  quanti- 
ties, the  root  of  the  equation  will  be  found  by  divid- 
ing both  members  by  Vie  coefficient  of  the  unknown 
quantity. 

Note.  If  the  coefficient  is  unity,  there  will,  of  course,  be  no  need 
of  dividing. 

§  48.  Bringing  together  the  principles  above  explained 
(§§  43-47),  we  have,  for  solving  equations  of  the  first  de- 
gree, containing  but  one  unknown  quantity,  the  following 


RULE. 

Clear  the  equation  of  fractions,  and  bring  alt  the. 
terms  containing'  the  unknown  quantity  upon  one  side, 
and  all  the  known  terms  upon  the  other.     Reduce  tht 
two  members  to  their  simplest  form,  and  divide  them 
both  by  the  coefficient  of  the  unknown  quantity. 

1.  Given  -  +2  =  -  -|-  -  -|-3,  to  find  x.    Ans.  x  =  20, 

2.  Given  6|+^-3  =  ^+^+2H,  to  find  *. 

6£  and  2\^  are  obviously  the  same  as  6-J-£  and  2-f-}£. 
Either  form  may  be  used.  In  this  instance,  the  latter  form 
will  be  found  more  convenient  for  reduction. 

3.  Given  x—%xz=33—3x,  to  find  x. 

*5 


54  INTRODUCTION.  [§49-51. 

§  49.  Many  equations,  which  are  not  of  the  jirst  degree. 
can  be  so  easily  reduced  to  that  form,  that  they  may,  pro- 
perly enough,  be  briefly  considered  in  this  place. 

I.  An  equation  may  contain  higher  or  lower  powers  of 
the  unknown  quantity,  which  may  be  canceled  by  transpo- 
sition, so  as  to  leave  no  power  higher  than  the  first,  or  low- 
er than  the  zero  power.     Thus, 

Let  *"+?  +  -£  +3  =  x+af. 

Canceling  a",  we  have 

— -| — = — [-3  =  x,  an  equation  of  the  first  degree, 
o       o 

Equations  of  this  form,  or  which,  on  reduction,  take  this 

form,  need  no  farther  remark. 

§  50.  II.  An  equation  may  contain  only  the  zero  and  mi- 
nus first  powers  of  the  unknown  quantity.  This  may  pro- 
perly be  called  an  equation  of  the  minus  first  degree.  But, 
if  we  multiply  by  the  unknown  quantity,  we  shall  evident- 
ly reduce  the  equation  to  the  common  form  of  the  first  de- 
gree.    Thus, 

Let  x-l+2x~1+3x-i  =  2: 

or  6a;-1  —  2. 

Multiply  by  x ; 
then  6x°  —  2x;  or  &=2x.  <  12.  - 

x=-2>. 

Otherwise,  --j [--  =  2.  '< 

xx      x 

Clearing  of  fractions,  1+2+3  =  2x ;  or  6  =  2x.       §46. 

x  —  o. 

Hence, 

Equations  of  the  minus  first  degree  can  be  reduced  to  the 
first  degree  by  multiplying  by  the  unknown  quantity. 

§51.  III.  Any  equation  containing  only  tico  powers  of 
the  unknown  quantity,  provided  their  exponents  differ  by 


52.]  EQUATIONS.  00 

unity,  may  evidently  be  reduced  to  the  common  form  of  the 
first  degree, by  dividing  by  the  lowest  poicer  of  the  unknown 
quantity.     Thus, 

Let  x2— lOx  =  0  ;  orx2  =  10./ . 

Divide  by  x ; 
then  a;— 10  =  0;  or  x  =10.  §  42.  c£ 

So  also  if  .Tn=5a:n-i, 

then,  dividing  by  a.-"-1,     x  —  5. 

1.  Given  3x5-f-2a;4 — x3  =  lx5-{-llx'1,  to  find  x. 

3  1  3  X 

2.  Given  x?-\-2x'-  =  ±x2-\-5x2,  to  find  a:. 

a.)  The  principle  of  §  51  obviously  includes  that  of  §  50. 
inasmuch  as  dividing  by  x~1  is  the  same  as  multiplying  by 

h.)  The  whole  class  of  equations  included  under  §  51,  are 
actually  of  the  first  degree,  according  to  the  more  general 
definition  of  the  degree  of  an  equation.  For,  the  range  of 
the  degrees  of  the  terms  with  respect  to  the  unknown  quan- 
tity, from  loicest  to  highest,  is  expressed  by  unity  (§  40.  a)  ; 
as  is  found  by  subtracting  the  lowest  from  the  highest. 

§  52.  IV.  An  equation  may  contain,  besides  the  zero 
power  of  the  unknown  quantity,  only  a  simple  root  of  the 
unknown  quantity ;  i.  e.  it  may  contain  only  the  zero  and 
the  one   half,  one  third,  or  ith  powers   of  the   unknown 

quantity.     The  equation,  in  this  case,  is  of  the  one  half, 
one  third,  or  Ith  degree.     Thus, 

\  Let  x  =  5  ;  or^Ac  — 5. 

Squaring  both  members  (§  42.  c), 
we  have  x  =  25. 

So,  if  we  had  xn  —  a,  or  n*/x  =z  a,  we  should  find  x  =  a". 
Hence, 

An  equation  of  the  1th  degree  can  be  reduced  to  tbe  first 
degree,  by  raising  both  members  to  the  nth  power. 


56  INTRODUCTION.  [§  53-55. 

Note.  This  operation  evidently  comes  under  §42.  c,  for  the 
members  being  equal,  multiplying  them  by  themselves  is  multiply- 
ing them  by  equal  quantities.  So,  if  they  be  separated  into  the  same 
number  of  equal  factors,  one  factor  on  one  side  will  be  equal  to  one 
on  the  other;  i.  e.  any  fractional  power  or  root  of  one  side  is  equal 
to  the  same  power  or  root  of  the  other.  For  this  is  formed  by  divid- 
ing all  the  factors  but  one  out  of  each  member  (§  42.  d).     Hence, 

If  both  members  of  an  equation  be  raised  to  the  same  power, 
whether  integral  or  fractional  (§22),  the  results  will  be  equal. 
i  i 

1.  Given  Jar3+2  =  ^-\-o,  to  find  x. 

2.  Given  y4+2  z=  -^-  -f-8,  to  find  y. 

o 

§  53.  We  have  classed  equations  with  reference  to  their 
unknown  quantities.  They  are  also  sometimes  distinguish- 
ed, with  reference  to  the  form  in  which  their  hioion  quan- 
tities are  expressed,  as  numerical  or  literal. 

A  numerical  equation  is  one,  in  which  the  known  quanti- 
ties are  all  expressed  by  numbers  ;  as  x-  ■=.  10x-(-24. 

A  literal  equation  is  one,  in  which  a  part  or  all  of  the 
known  quantities  are  expressed  by  letters  ;  as  ax2-{-'2bx=.  c. 

§  54.  A  conditional  equation  is  the  algebraic  expression 
of  a  problem™  ;  i.  e.  something  proposed  to  be  performed 
or  discovered. 

Thus,  the  equation  x — 3  =  7  (§  43.  1),  proposes  this  prob- 
lem ;  viz.  To  find  a  number  such  that  if  it  be  diminished  by 
3,  the  remainder  shall  be  7. 

So  the  equation  Ja:-|-3  =  8  (§45.  1),  proposes  this  prob- 
lem; viz.  To  find  a  number,  whose  fourth  part,  increased 
by  3,  is  equal  to  8. 

State,  in  like  manner,  the  problems  involved  in  each  of 
the  equations  of  §§  43-52.     Compare  §  3.  a. 

§55.  As  we  bave  seen,  an  equation  is  the  algebraic  ex- 
pression of  a  problem ;  and  the  solution  of  the  equation 
gives  the  solution  of  the  problem.     Hence  to  solve  a  proL- 

(w)  Gr.  npoffiriua,  from  Trpoj3u?J,<j,  to  throw  or  lay  before. 


§  00.]  PROBLEMS.  57 

km,  we  have  only  to  express  its  conditions  algebraically  by 
an  equation,  and  then  solve  the  equation. 

The  process  of  expressing  the  conditions  of  a  problem  by 
an  equation  is  sometimes  called  putting  the  problem  into  an 
equation  ;  and  is  frequently  more  difficult  than  the  subse- 
quent solution  of  the  equation. 

The  student  will  most  readily  learn  the  methods  of  form- 
ing an  equation  from  a  problem,  by  stating  the  problems 
involved  in  the  preceding  equations,  and  observing  how  the 
conditions  of  each  problem  are  expressed  in  the  equation. 
He  will  find,  that  the  process  conforms,  in  general,  to  th» 
following 

RULE. 

Represent  the  unknown  quantity  by  some  letter,  as 
x ;  then  combine  the  known  and  unknown  quantities 
according"  to  the  conditions  of  the  problem.  The  re- 
sult will  be  an  equation  expressing  those  conditions. 
See  §  3.  b. 

In  this  process,  we  treat  the  unknown  quantity  as  if  it 
were  known ;  and  perform  upon  it  just  those  operations 
which  would  be  necessary  to  prove  the  correetnesss  of  the 
result,  if  we  had  fixed  upon  a  value  for  the  unknown  quan- 
tity. We  have,  in  fact,  fixed  upon  a  representative  of  that 
value,  in  the  letter  which  we  have  chosen  to  denote  the  un- 
known quantity. 

Problem  1.  To  find  a  number  whose  fifth  part  exceed- 
its  sixth  part  by  10. 

Let  x  represent  the  number  sought. 

r 

Then  ±x,  or  "-  will  represent  its  fifth  part. 
o 

X 

and  \x,  or  -  will  represent  its  sixth  part. 
Then,  by  the  condition, 

^  —  -10. 

0       6 


58 


INTRODUCTION.  [§  00. 


6a;— 5a;  =  300.  §46. 

or  x  =  300. 

Verification,       —  _  ££2  —  60-50  =  10. 
5  6 

Prob.  2.  What  sum  of  money  is  that,  whose  fourth  part 
exceeds  its  fifth  part  by  5  dollars  ? 

Prob.  3.  What  sum  of  money  is  that,  whose  fourth  part 
exceeds  its  fifth  part  by  a  dollars  ? 

Let  x  =  the  sum. 

Then  *     %  —  a. 

4      5 

•  ••  (  §46)  5x — 4x  =  20a ;  or  x  =  20a,  the  sum. 

Note.  The  last  problem,  it  will  be  seen,  is  the  same  as  the  pre- 
ceding, except  that  the  difference  between  the  fourth  and  fifth  parts 
of  the  number  is  denoted  by  a,  which  may  represent  any  number 
whatever.  This  is  called  a  general  solution,  or  generalization  of 
the  problem.  In  this  solution,  a,  the  given  excess  of  the  fourth  p;;rt 
above  the  fifth,  remains  in  the  result;  whence  we  learn,  that  the 
whole  number  must  be  20  times  that  excess.  Thus,  if  that  excess 
be  1,  the  number  must  be  20;  if  the  excess  be  2,  the  number  must 
be  40;  if  the  excess  be  5,  the  number  will  be  100;  &c. 

Prob.  4.  A,  B,  and  C  enter  into  partnership.  A  con- 
tributes a  certain  sum ;  B  contributes  three  times,  and  C, 
four  times  as  much  as  A.  Their  whole  stock  is  $20,000, 
How  much  did  each  contribute  ? 

Let  x  =  A's  part ;  then  3x  =  B's,  and  4x  =  C's. 
x-\-3x+4x  =  20,000. 

Prob.  8.  A  man  and  boy  work  together,  for  S75.  The 
man's  work  is  worth  four  times  as  much  as  the  boy's.  How 
shall  they  divide  the  money? 


CHAPTER  I. 


ADDITION  AND  SUBTRACTION. 


I.  ADDITION. 

§  56.  Addition  is  the  process  of  finding  the  aggre- 
gate of  several  quantities.     See  §  8. 

Adding  quantities  is  bringing  them  together,  so  that  each 
may  have  its  proper  effect  in  making  up  the  aggregate ; 
those  which  increase,  and  those  which  diminish  the  amount, 
being  characterized,  each  by  the  proper  sign.  See  §  4. 
Hence,  for  adding  quantities,  we  have  the  following 

RULE. 

§  57.  Write  the  quantities  to  be  added,  one  after 
another,  each  with  its  own  sign. 

a.)  If  the  polynomial,  thus  formed,  contain  similar  terms, 
it  may,  of  course,  be  reduced  by  §  34. 

b.)  This  reduction  can  often  be  easily  performed  without 
first  writing  out  all  the  terms  at  full  length.  For  this  pur- 
pose, there  is  an  advantage  in  writing  the  similar  terms 
under  one  another.     Thus, 

Add  a*+2a3y-\-a*y2,  —  2a3y— 4a2y2— 2ay3,  and  a2y2 
+2ay*+y*. 


60  ADDITION.  [§  57. 

Writing  these  expressions  with  their  similar  terms  un- 
<1er  one  another,  we  have, 

a*  +  2a3y  +   a*y* 

—  2a3y  —  Aa2y2  —  2ay3 

«y  +  2«y3+y4 

a4  —  2a2y  +y4 

1.  Add  a-\-b  to  c-\-x.  Sum,  a-\-b-\-c-\-x. 

2.  Add  a-\-b,  and  a — b.  Sum,  2a. 

3.  "     Ja-f-^J,  and  \a — \b.     Sum,  a.     That  is,  the  half 
sum  -f-  the  half  difference  of  any  two  quantities  ==  the  greater. 

4.  Add  a2-\-ab,  and  aJ-(-52. 

5.  "     a" — ab,  and  ab — b2. 

6.  "     a3— 2a2&-j-«&2,  and  o26— 2ai24-£3. 

7.  "     a;3-|-2^2^+«y2,  and  ar^-f-Sx'y2-^3- 

8.  "     y2-\-x2 — px-\-\p2,  and  — x2 — px — \p2. 
c-x2        ,    .       „        c2x2 


9.  «     ^2+2cxH j— ,  and  A2—2cx-\ 

\        ]     A2'  l    A2 

10.  «    y2-(-^24-2cx+r2,  and  y2+x2— 2c«-f-c2« 

o™2  o*3  1^4  "y»5  'y*  6  'y7  t>8  qcO 

"•  "    ^-T+T-T  +  T-T+T-T  +  IT' 

^»  2  ^y»3  T*^  9"  ^  /}r>®  ^/*^  -1*8  CP^ 

and  -^-T-T""T_T— 6—  y— s~ T 

12.  Add  a2-j-2a5-fJ2,  and  a2—  2ab+b2. 

13.  "     sin  a  cos  £>-|-cos  a  sin  b,  to  sin  a  cos  b — cos  a  sin  b. 

14.  "     2a  +  a2x-1,    8a*~x ""*,   6aar°,   lOa5^,  — 15a°a:, 

— 12a~-x^,  9a-1;*;2,  lOa2^-1,  lla^aP*,  8«ar°? 

1      3 

— 5^/a^/x-|-5a0a:-f-2a  2x5,  and  18a0x. 


15.     "     ay-\-bx,  and  a'y — J'.t. 


+v 


Ans.  (a-\-a')y-\-(b — V)x,  or  a 

+a>' 

1  G.    "     ay — bx-\-cz,  a'y-\-Vx — c'z,  and  — a"y-\-b"x — c"z. 

17.  "  y3-\-ay2-\-aby ;  by2-\-acy,  and  cy2-\-bcy-\-abc ; 
and  arrange  the  result  according  to  the  descending  powers 
ofy  (§§33,  34.  c). 


§  58.]  PROBLEMS.  61 

18.   Add,  member  by  member  (Geom.  §  22),  the  equa- 
tions —7x-\-5y  =  19,  and  10a;— 5y  =  — 10. 

Arts.  3#—  9.   .-.  x  =  what? 


PROBLEMS. 

§  58.  1.  The  sum  of  2x— 10  and  4x— 20  is  equal  to  3a;. 
What  is  the  value  of  x  ? 

2x— 10+4»— 20  =  3x. 
Qx— 30  =  3x.   .-.  6x— 3a;  =  30  ; 
or  Sx  =  30.   .-.  x  =  10. 

2.  The  sum  of  5a; — 8,  2x — 20,  and  x — 10  is  equal  to 
10 — 4a;.     What  is  the  value  of  x  ? 

3.  The  sum  of  %x — 1,  2 — | x,  l-\-x — §x,  and  x — 2  is  equal 
to  ar-f-5.     What  is  the  value  of  x  ? 

4.  The  sum  of  2x,  7x,  fa:,  and  — 6  is  — 23.  What  is 
the  value  of  x  ? 

5.  The  sum  of  13| — \x  and  — 2a?4-8§  is  nothing.  What 
is  the  value  of  a;? 

13|— %x— 2x-f8|  =  0.  .-.  22£ z=  2\x.  .-.  x=  9. 

6.  A's  property  is  3a  dollars,  and  his  debts  2a ;    B's 

property  is  5a,  and  his  debts  3a ;   if  they  make  common 

stock  of  their  property,  what  is  their  net  capital,  x  ? 

Let  a  =  100,  500,  1000, 10,000,  and  find  the  value  of  x  in  each 
case. 

7.  An  estate  was  divided  among  three  sons.  The  eld- 
est received  $4000  less  than  one  half;  the  second  received 
one  third ;  and  the  youngest  received  S2000  more  than  one 
quarter  of  the  whole.  What  was  the  estate,  and  what  did 
each  receive  ? 

Let  the  estate  be  represented  by  x.    Then  we  shall  have 

the  share  of  the  first  —^  —  4000  ; 

x 
"  second  =  -,  and 

o 

ALG.  6 


62  SUBTRACTION.  [§  59. 

share  of  the  third  =  -  -j-2000.     The  sum  of  the  shares 

's,  of  course,  equal  to  x. 

8.  Let  the  first  receive  a  less  than  half;  the  second,  one 
third ;  and  the  youngest,  one  half  of  a  more  than  one  quar- 
ter of  the  estate.  What  was  the  estate,  and  what  did  each 
receive  ? 

3C  3C  CC        Ct 

Here  the  shares  are  - — a,  -,  and  j  +  <j' 

x  =  6a,  the  estate. 
Let  a  —  1000,  100,  2000,  10,000,  and  find  the  value  of  the  estate, 
and  the  share  of  each. 

9.  A,  B  and  C,  form  a  partnership ;  A  puts  in  a  cer- 
tain amount  of  stock;  B  puis  in  $2000  less  than  the  dou- 
ble of  A's ;  and  C  invests  $8000  less  than  the  triple  of  A's. 
The  whole  stock  is  $50,000.     Required  each  one's  share. 

10.  Suppose  the  sum  of  the  distances  of  Mercury,  Ve- 
nus, and  the  Earth  from  the  Sun  is,  in  round  numbers, 
200  millions  of  miles ;  and  that  the  distance  of  Mercury  is 
31  millions  less,  and  of  the  Earth  58  millions  more  than 
that  of  Venus.     What  are  their  several  distances  ? 

Let  xz=z  the  distance  of  Venus. 


II.  SUBTRACTION. 

§  59.  Subtraction  is  the  process  of  finding  the 
difference  between  two  quantities.     See  §  8. 

a.)  We  have  seen  (§  7.  b),  that  the  subtraction  of  a  neg- 
ative quantity  has  the  same  effect  as  the  addition  of  an 
equal  positive  quantity.  Therefore,  to  subtract  a  negative 
quantity,  we  have  only  to  change  its  sign  and  add  it. 


B  gO.1  SUBTRACTION.  63 

h.)  It  is  also  perfectly  obvious,  that,  to  subtract  a  posi- 
tive quantity,  we  have  only  to  put  the  sign  —  before  it ; 
i.  e.  to  change  its  sign,  and  add  it. 

Hence,  for  subtracting  one  quantity  from  another,  we 
have  the  following 

RULE. 

§  60.  Change  the  signs  of  the  quantity  to  be  sub- 
tracted, and  then  add  the  two  quantities. 

a.)  In  expressions  of  this  form,  a+(b— c)— (c-f-#)+ 
(a_j_&)(c— x),  the  quantities  enclosed  in  the  symbol  of  union 
(§  2.  h),  as  (b—c),  (c— y),  and  such  quantities  multiplied  to- 
gether, as  (a+*)(e— x),  are  to  be  regarded  as  single  com- 
pound, or  complex  terms  ;  and  the  rule  applies  to  the  sign 
before  the  whole  term,  and  not  to  the  signs  between  the 
parts  of  the  term.  Thus,  to  subtract  a—  (b— c)+(«— b) 
__(a_g)(_c)  from  ff,  we  write 

g—a-\-(b—c)—(a—b)+(a-b)(—c). 

Here,  the  addition  of  (b—c),  and  the  subtraction  of  (a— b) 
is  indicated.  If  this  addition  and  subtraction  also  be  per- 
formed, we  shall  have  g— a-{-b— c— a-\-b-\- (a— b)(— c). 

Note.  An  operation  is  said  to  be  indicated,  when,  without  be- 
ing actually  performed,  it  is  denoted  by  the  proper  symbol.  Thus, 
3abX'2ab  is  an  indicated  multiplication.  So  subtraction  is  indicated 
by  writing  the  subtrahend'  in  a  parenthesis,  and  placing  the  sign  — 
before  it.     Thus  a — {b — c). 

b.)  In  subtracting  a  term  preceded  by  the  double  sign. 
the  order  of  the  signs  will  obviously  be  inverted.  Thus, 
a — (±b)  =  a^fb;  i.  e.  plus  or  minus  is  changed  to  minus  or 
pins.     10— (±5)=  10^:5  =%  or  15. 

1.  From  a,  subtract  b-\-c.  Remainder,  a — b — c. 

2.  From  a-\-b,  subtract  c.  Rem.  a-\-b — c. 

3.  a+b—  (a— J)  =  what?  Am.  2b. 

(x)  Lat.  subtrahendus,  to  be  subtracted,  from  eubtraho  to  take 
away,  subtract. 


64  SUBTRACTION.  [§61. 

4.  lx-\-Ly—  (\x— \y)  =.  what  ?  Am.  y.  That  is,  the 
half  sum  —  the  half  difference  of  any  two  quantities  =  the 
less. 

5.  From  x2—y2,  subtract  x*-\-xy.       Rem.  — xy—y2. 

6.  From  an—bn,  subtract  an—ban-1. 

7.  From  y2-{-x2-\-2cx-\-c2,  subtract  y3-f-x2— 2cx-\-c2. 

8.  From  A2+2cx  \  C*~*\  take  A*— 2ex  1  °*f . 

A*  ■  '   J.2 

9.  From  a3 — 53,  take  «3 — bet2. 

10.  From  ba2—b3,  take  a2b—ab2. 

11.  From  ab2—b3,  take  ab2—b3. 

12.  From   3&23ri+4&V~*— 6fy°— 10&V+8S"^  take 

— 2&V1— %°-f2&V^+5^"—  26°y+3&~^—  3/>-^2. 

#2      x3      ic*  a?2       e3 

13.  From   x —  -4— p    subtract    — .r — — 

a:4 

T 

14.  From  1  Ox— ly  —  30,  subtract  8a:— ly  =  20. 

15.  From^V-he^2^:^2^2,  subtract  ^V2+^2 
x"2=A2B2.        Rem.  A2{y2—y"2)+B2{x-—x"2)  =  0. 

§61.  It  is  sometimes  important  to  indicate  the  subtrac- 
tion of  a  polynomial,  without  actually  performing  it. 

Thus,  a-\-x — (b-\-c — d)  which,  when   performed,   : 
a-\-x — b — c-\-d. 

As,  in  performing  a  subtraction  which  has  been  ind 
ed,  we  change  all  the  signs  of  the  quantity  within  the  pa- 
renthesis ;  so  we  may  return  from  a  performed,  to  an  indi- 
cated subtraction,  by  re-changing  all  the  signs  of  the  quan- 
tity whose  subtraction  is  to  be  indicated,  and  enclosing  the 
terms  hi  a  parenthesis,  with  the  sign  —  before  it.  We 
may,  therefore,  put  a  polynomial  under  different  forms, 
without  affecting  its  value.     Thus, 

J2_2Jc_j_c2  —  J2_(2JC— c2)  =  —(—b2-\-2bc—C'  ) . 

cf—A2-\-B2  =  cy"—{A 2—B2 ) . 


§  62,  63.]  COMBINATION  OF  SIGNS.  65 

1.  _ (a:2—  _42)  =  what?     R2— (cos  «coa  b— sin  a  sini)? 

2.  Indicate,  in  every  way  possible,  without  changing  the 
order  of  the  terms,  the  subtraction  of  r—s-\-t—u  from  a. 

§  62.  We  have  already  found,  in  several  instances,  two 
signs  combined  before  a  single  term  (§§7.  a,  b,  60.  b). 
There  is  nothing  to  hinder  any  number  of  signs  from  being 
thus  combined.  It  is  proper  therefore  to  consider  the  ef- 
fect of  such  a  combination. 

a.)  In  the  first  place,  as  addition  is  simply  the  bringing 
of  quantities  together  in  their  proper  character  (§  54),  the 
sign  -\-  can  never  change  the  previously  existing  sign  of  a 
term.  "Whether  employed  once  or  oftener,  it  simply  leaves 
the  sign  of  the  term  as  it  was  before  the  sign  -f-  was  pre- 
fixed.    Thus, 

a-\-( — b)  [i.  e.  a  together  with  — 5]  =  a — b. 

Hence,  in  estimating  the  effect  of  any  number  of  signs, 
the  positive  signs  may  be  disregarded ;  the  sign  of  the  term 
Upends  upon  the  negative  signs. 

b.)    As  subtraction,  on  the  other  hand,  always  changes 
the  sign  of  a  term,  the  sign  —  always  reverses  the  charac- 
ter of  the  term  to  which  it  is  prefixed.     Thus, 
-\-a  =  a  (§4.  a);    .-.  — (+«)  =  — «• 

Again  — ( — a)  =  -\-a ;  §  7.  a,  b. 

—(—(—«)  =  — (+a)  ——a. 

Hence, 

§  63.  If  the  number  of  negative  signs  before  a  term 
be  evexN,  the  resulting  sign  is  +;  if  odd, — .  Com- 
pare §11.  Note  2. 

Note.  This  includes  the  case,  in  which  the  signs  are  all  positive. 
For  then  the  number  of  negative  signs  is  represented  by  0,  an  even 
number,  being  less  by  unity  than  1,  which  is  an  odd  number. 

1.   What  is  the  value  of  —  (4-'—  26c+c2)  ? 

Ans.  —  (b*— 2bc+c*)  =— (&»)— (— 2bc)— (+c3)  ——{,< 

-\-2bc— c2. 

*6 


66  SUBTRACTION.  [§  64,  65. 

2.  What  is  the  proper  sign  of — ( — «2)?  of — ( — ( — a3)? 
<rf— (—(—(— «4)?  of+(+a)?  of+(+(+a)?  of+(— a)' 
of+(-(-a)?    of  +(-(+«)? 


PROBLEMS. 

§  64.  1.   The  remainder  found  by  subtracting  — 23 — \x 
from  2x-\-l  is  6a;.     What  is  the  value  of  x  ? 

2.   The  difference  between  8a; — 5  and  — 7x-(-12  is  noth- 
ing.    What  is  the  value  of  x  ? 

8x— 5— (— 7x+12)  =  0. 
3a;      7x      /7x      3x\ 

T~"io" 

4.   Aa—Av—{Bb+Bv)=0.    v  =  what? 

Aa—Bh 


n      ~~         .  ~         flX        OX\  ,  „  , 

-(—  —  —  )=  — 15.     a;  =  what.-' 


Am-V~   A+B 
5.    Divide  54  into  two  such  parts,  that  the  less  subtract- 
ed from  the  greater,  minus  the  greater  subtracted  from 
three  times  the  less,  shall  be  equal  to  nothing. 

Let  x  =■  the  less  ; 

then  54 — x  =.  the  greater. 

54— x— x—  (3a;—  (54— x))  =  0. 

§  65.    1.  A  is  10  years  older  than  B,  and  the  sum  of 
their  ages  is  60.     What  are  their  ages  ? 
Let  x  —  A's  age  ; 

Then  x— 10  =  B's  age, 

and  x-\-x— 10  =  the  sum  of  their  ages,  which  is  60. 

x+x— 10  =  60. 

2a;  =  70.  §§34,44. 

x  =  35,  A's  age, 
and  x— 10=  25,  B's  age. 

Or,  let  x  =  B's  age. 

Then  x+1 0  =  A's  age,  &c 


65.]  PROBLEMS. 

( )r,  again,  let  x  =  A's  age  ; 

and  60 — a::=B'sage. 

Then  x — (60 — x)  =  the  difference  of  their  a,u>  - 

which  is  10. 

x—  (60— x)  =  10. 

2.  The  sum  of  two  numbers  is  100,  and  their  difference 
is  20.     What  are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  S,  and  their  difference 

i*  D.     What  are  the  numbers  ? 

ci_  n 
Ans.    The  greater  is  — - — ,or    hS-\~hD,  and   the    less, 

-^-,  or  }&-*!>. 

Note.  The  1st,  2d,  and  3d  examples  propose  the  same  question 
under  different  forma.  But,  in  the  3d,  the  quantities  employed  re- 
main in  the  result  (§  55.  3.  N.),  and  show  how  they  are  employed 
to  obtain  that  result.  Thus  S  denotes  the  sum  of  any  two  numbeis. 
and  X),  their  difference ;  and  we  find  the  greater  by  adding  the  differ- 
ence to  the  sum,  and  dividing  by  two;  and  the  less,  by  subtracting 
the  difference  from  the  sum,  and  dividing  by  two.  (Compare  §  57. 
2,  3,  §60.  3,  4,  and  Geom.  §22.)     Thus, 

Let  the  sum  of  two  numbers  be  50,  and  their  difference, 
6  ;  and  find  the  numbers.     Here  #=50,  and  D  =  G  : 

S+D      56      no       3  S-£>      44      an 
^  =  T  =  28,and—  =T  =  22. 

And  we  find     28+22  =50  =  S,  the  sum  ; 

and  28—22=    6  =  D,  the  difference. 

Let  the  sum  be  75,  and  difference  25  ; 
12,  "         2; 

"  12,  «         3; 

"         "  19  «        7  • 

«        «  75°27',         "         13°15' 


what  are 
►  the  num- 
bers? 


CHAPTER  II. 


MULTIPLICATION  AND  DIVISION. 


I.  MULTIPLICATION. 

§  66.  Multiplication  is  the  process  of  combining 
factors  into  a  product  (see  §  10) ;  in  other  words,  it 
is  the  process  of  taking  as  a  term,  one  quantity  called 
the  multiplicand10,  as  many  times  or  parts  of  a  time, 
as  there  are  units  or  parts  of  a  unit,  in  another  quan- 
tity called  the  multiplier. 

Thus,  if  6  dollars  be  taken  as  a  term  3  times,  the  result 

is  6x3  =  6— f— 6— f— 6  =■  18  ;  if  6  dollars  be  taken  as  a  term  § 

of  a  time,  the  result  is  6X|  =  (2+2+2)§  =  2+2  =  4. 

Note.  It  is  obvious,  that,  in  numbers,  either  factor  may  be  made 
the  multiplicand,  and  the  other,  the  multiplier,  without  affecting  the 
result.     See  Geom.  §  172. 


MULTIPLICATION  OF  MONOMIALS. 

§  67.  All  multiplication  resolves  itself,  as  we  shall  see, 
into  the  multiplication  of  monomials.  We  shall,  therefore, 
consider  that  case  first. 

Numerical  coefficients  are,  of  course,  subject  to  the  prin- 
ciples of  Arithmetic,  and  must  be  multiplied  accordingly. 
Letters,  we  hare  seen,  are  multiplied  by  writing  them  to- 
gether (§  2.  e.  N.)  ;  and  powers,  by  adding  their  exponents 

(w)  Lat.  multiplicandus,  to  be  multiplied,  from  multiplico,  com- 
pounded of  multus,  many,  and  plico,  to  fold ;  as  if  the  quantity  were 
folded  on,  or  added  to,  itself. 


§  68.]  MULTIPLICATION  OF  MONOMIALS.  69 

i    24.  a).     Hence,  we  have,  for  the  multiplication  of  mono- 
mials, the  following 

RULE. 

§  68.  Multiply  the  numerical  coefficients  as  in  Arith- 
metic; and  annex  the  letters  of  the  factors,  giving-  to 
each  an  exponent  equal  to  the  sum  of  its  exponents  in 
the  factors. 

a.)  We  have  shown  (§  9.  a),  that  the  product  of  two  fa<  - 
tors  of  like  signs  is  positive,  and  of  unlike  signs,  negative ; 
and  (§11.  N.  2),  that  the  product  of  any  even  number  of 
negative  factors  is  positive,  and  of  any  odd  number,  negative. 
We  have  also  shown  (§  62.  a),  that  positive  signs  have  no 
effect  to  change  the  sign  of  a  term ;  but  that  the  sign  de- 
pends upon  the  negative  signs.  Hence,  whatever  be  th< 
number  of  factors, 

If  the  number  of  negative  factors  be  even,  the  product 
is  positive  ;  if  odd,  negative. 

b.)  The  sign  of  each  factor  obviously  produces  its  effect 
upon  the  whole  product  (§  9.  a).  Hence,  we  may  write  the 
signs  of  all  the  factors  before  the  product,  and  determine 
the  resulting  sign  by  §  63. 

c.)  When  one  only  of  the  factors  has  a  double  sign  (±  or 
:f ),  the  sign  of  the  product  will,  of  course,  be  double ;  and 
will  be  either  the  same  as  that  of  the  factor,  or  inverted, 
according  as  an  even  or  odd  number  of  the  remaining  fac- 
tors may  be  negative.     Thus, 

±aXb=±ab;  ±aX — b=^:ab;  ^:abX — c  —  ±abc. 
(±o)  ( — b) ( — c)  —  ±abc  ;  ±a. — b. — c. — x  =  ^abcx. 

if  two  factors  have  each  a  double  sign,  and  if  it  be  un- 
derstood, that  the  upper  signs  must  be  taken  together,  and 
the  lower  signs  together,  the  sign  of  the  product  will,  obvi- 
ously,  be  single;  and,  if  the  signs  of  the  factors  be  alike, 
the  product  will  be  positive  ;  if  unlike,  negative.  Thus, 
±'>X±b  =  -\-b;  ±aXTb  =  —'-d.     [±a)[±b){^:c)—  qpaSe. 


7(1  MULTIPLICATION.  [§  69,  7<». 

d.)  The  degree  of  the  product  of  several  monomial  fac- 
tors is,  evidently,  equal  to  the  sum  of  the  degrees  of  those 
factors  (28). 

1.  Multiply  together  2a-b,  —3ab'2,  Aa~lb~2,  and  —\b. 

Product  12«  -'/;•-'. 

2.  SaX—bX—cX—2hy  =  what  ? 

3 .  a'"  X  a-"  X  b"  X« Jr 1  =  what  ?  Jns.  a"-"+ *  6"  - J . 

4.  Multiply  together  |,  — J,  i?-3,  — a;2,  and  — x2. 

5.  (±a#X±#)  — what?     ax2(±x)?     ( — «x3)(±.r)  r 

MULTIPLICATION  OF  POLYNOMIALS. 

■  69.  First,  let  one  factor  only  be  a  polynomial.     Thus, 

Multiply  together  b-\-y  and  a. 

(b-\-y)  times  a  is  the  same  as  a  times  (b-\-y)  [see  §  66, 
X.]  ;   i.  e.  a  times  the  sum  of  b  and  y ;  which  is,  obviously, 
the  same  as  the  sum  of  a  times  b,  and  a  times  y. 
(b-\-y)a,  or  a (&-[-#)  =  «&-}-«#• 

Hence,  the  product  of  a  polynomial  into  a  monomial  con- 
sists of  the  aggregate  of  the  products  of  the  monomial  into 
the  several  terms  of  the  polynomial.     See  Geom.  §  178.  1. 

1.  Multiply  a2— 2ab+b*  by  a. 

Prod  a3—2a*b-\-ab*. 

2.  (a  2  ±  2«/;+6 2)  X  ± b  =  what  ? 

.4ms.  ±aVj-\-2ab2±l 

3.  (j2_L^2_a2)X_J22  — what? 

4.  Multiply  l+Ja-2u;2— |o-***+A«-8a:e  by  a. 

§  70.   Again  let  there  be  two  polynomial  factors.     Thus. 
Multiply  a-\-b  by  c-\-y. 

(c-j-y)  times  a-\-b  is  evidently  the  same  as  c  times  cir\-b, 
added  to  y  times  a-\-b ;  i.  e. 

[a-\-b)[c-\-y)  =  («-f-%-R«+%  =  ac-j-Je-f-ay+Jy.       S.  i 

§67. 


§71.] 


OF   POLYNOMIALS. 


i 


Hence,  we  have,  for  the  multiplication  of  polynomial- 


the  following 


RULE. 


§  71.  Multiply  each  term  of  the  multiplicand  by  each 
term  of  the  multiplier,  and  add  the  products.  See 
Geom.  §  178.  Cor.  III. 

a.)    This  is  precisely  the  method  employed  in  Arithme- 
tic.    Thus,  to  multiply  84  by  25,  we  have 
34  =  30+  4 

25=  20+  5 

170=  150+2o 

68    =    600+80 


850=    600+230+20 

1.  (a2+2b)(a~2— b2)  =  what  ?    (a+b){a+b)  =  (a+b)2  ? 

a2+2b  a+b 

a-2—b*  a+b 

a"+2a~2b  a2+ab 

—an2— -2b3  ab+b2 

l+2a~2b—a2b2—2b3  a2+2ab+b2 

2.  ^+ay+y*)(a2-ay+y*)  =  what? 

Ans.  a*+a2y2+y±. 
(a*—b2)  (a—b)  =  what  ?    yircs.  a3— a2b— ab2+b3. 
{a+b)(a—b)(a—b)  =  what  ?     («+£>)(«— 6)(a+6)  ? 
{a3+3a2b+Sab^+b3)[a2+2ab+b2)  —  what  ? 
(2a4—' 3«2Z>2+4&4)(x— ^/)2x  =  what  ? 

{a+b—c)(a—b+c)  =  what  ?     (a-+^)(a;*— y*)  ? 


o. 

4. 
5. 
6. 


8.  (x — a)(x-\-b)  =  what  ?  -4ras.  x2 — a 

9 .  (x+a)(x+b)[x — c)(x—e)  =  what  ? 

10.  (a2±6z2+cz3)(l±H-z2±z3)  =  what? 


cc — ab. 


Ans.  az±a 

z2+a 

z*+b 

±b 

+i 

±b 

+c 

+c 

±c 

z5±czr-. 


§  72-74.]  MULTIPLICATION.  72 

§  72.  If  two  polynomials  are  each  homogeneous,  their 
product  will  be  homogeneous  also.  For  the  degree  of  any 
term  in  the  product  is  equal  to  the  sum  of  the  degrees  of  a 
term  in  each  factor  (§  68.  d)  ;  and  those  degrees  being  the 
same  throughout,  their  sum  must  be  always  the  same,  and 
therefore  all  the  terms  of  the  product  will  be  of  the  same 
degree. 

Hence,  if,  in  multiplying  homogeneous  polynomials  to- 
gether, we  observe  that  the  degree  of  one  term  is  greater 
or  less  than  the  degree  of  the  other  terms,  we  may  know 
that  some  mistake  has  been  made. 

This  remark  is  the  more  important,  because  so  many  of 
the  investigations  of  Algebra,  especially  those  relating  to 
Geometry,  give  rise  to  homogeneous  expressions. 

§  73.  If  the  product  of  polynomials  contains  similar 
terms,  it  may,  of  course,  be  simplified  by  §  34.  But  it  is 
apparent  that,  if  the  factors  themselves  were  reduced  to 
their  simplest  form,  there  will  always  be  some  terms  of  the 
product  unlike  all  the  others,  and,  therefore,  incapable  of 
any  reduction  except  the  partial  reduction  explained  in 
§  34.  c.     These  are, 

1.  The  product  of  the  terms  containing  the  highest  powers 
of  any  letter,  in  each  of  the  factors  ;  and 

2.  The  product  of  the  terms  containing  the  lowest  pow- 
ers of  any  letter. 

For  these  two  terms  must  contain  that  letter,  the  one  with 
a  greater,  and  the  other  with  a  less  exponent,  than  any  of 
the  other  terms  or  partial  products ;  and,  consequently,  can- 
not be  similar  to  any  of  them.  Hence,  no  product,  involv- 
ing a  polynomial  factor,  can  consist  of  less  than  two  terms. 

§  74.  If  there  are  no  similar  terms  in  the  product  of  two 
polynomials,  the  whole  number  of  terms  in  the  product  will 
be  equal  to  the  product  of  the  number  of  terms  in  the  mul- 
tiplicand by  the  number  of  terms  in  the  multiplier. 

For,  if  there  be  four  terms  in  the  multiplicand,  and  one 


§  75,  76.]  DETACHED  COEFFICIENTS.  73 

in  the  multiplier,  there  will  be  four  terms  in  the  product ; 
another  term  in  the  multiplier  will  give  another  four  terms 
in  the  product,  and  so  on. 

Also,  if  we  introduce  another  factor,  the  same  reasoning 
will  apply  to  the  product  of  this  factor  into  the  former  pro- 
duct. Hence,  in  general,  if  there  is  no  reduction,  the  num- 
ber of  terms  in  any  product  is  equal  to  the  continued  pro- 
duct of  all  the  numbers  of  the  terms  in  the  several  factors. 

§  75.  The  multiplication  of  polynomials  is  frequently  in- 
dicated, without  being  performed.     Thus, 

a(y+A)2  =  a(y*-\-2yh+h*)  =  ay2+2ayh-\-ah*. 
(P — ^)  X  (p — c)  ;  a-\-b — c .  a-\-c — b  ;  is(f  s — a). 
When  a  multiplication,  so  indicated,  is  performed,  the 
expression  is  sometimes  said  to  be  developed. 

MULTIPLICATION  BY  DETACHED  COEFFICIENTS. 

§  76.  In  multiplying  polynomials  arranged  according  to 
the  powers  of  any  common  letter  or  letters,  that  letter  or 
those  letters  may  be  omitted  in  the  operation,  and  the  pow- 
ers supplied  in  the  result ;  the  product  of  the  highest  or 
lowest  powers  being  placed  in  the  first  term,  and  the  pow- 
ers then  regularly  descending  or  ascending  through  all  the 
terms. 

This  is  called  multiplication  by  detached  coeffi- 
cients ;  and  will  be  best  explained  by  a  few  examples. 
Thus, 

To  multiply  x2+2.z+l  by  x2 — 2x+l,  we  write  the  co- 
efficients, and  multiply,  as  follows  : 
1+2+1 
1—2+1 
1+2+1 
—2—4—2 

1+2+1 
1+0 — 2+0+1.     Supplying  the  powers  of  x, 
we  have  a^+Oz3— 2z2+0a+l  =x4— 2x2+l. 

ALG.  7 


74  MULTIPLICATION.  [§  77. 

Multiply  «2+2a&+62  by  a-\-b. 

Here  the  polynomial  factors  being  arranged  with  respect 
to  both  the  letters,  both  may  be  omitted,  and  afterwards 
supplied,  one  with  descending,  the  other  with  ascending 
powers.     Thus, 

1+1 
1+2+1 

1+2+1 
1+3+3+1.     Supplying  the  letters, 

we  have  a3+3a2&+8«&2+&3. 

a.)  In  adding  the  coefficients  of  the  partial  products  in 
the  first  example,  we  obtain  zero  in  the  second  and  fourth 
places.  The  cypher  must  be  written,  to  occupy  the  place 
of  the  term,  and  show  what  powers  of  the  letters  fall  out. 
In  like  manner,  if  any  power  of  a  letter,  between  the  high- 
est and  lowest  in  any  factor,  be  wanting,  zero  should  be  re- 
garded as  its  coefficient,  and  written  in  its  place.  This  will 
fill  out  the  series,  aud  will,  obviously,  cause  the  coefficients 
of  similar  terms  to  stand  under  one  another.     Thus, 

3.   Multiply  a2+2a?/+#2  by  a2— y9. 
1+2+1 
1+0—1 
1+2+1 
0+0+0 
—1—2—1 


1+2+0—2—1. 
The  product  is  a4+2a3#— *2ay3— y*. 

4.  Multiply  z3— 3z2#+3z?/2— y*  by  z2— 2z#+#2. 

5.  (a+&)3  =  what?     (a+5)4?     (a+6)*? 
C.    03+z VH#2+3/3)  0— y)  =  what  ? 

PROBLEMS. 

§77.    1.    Given  *— i(2x+l)  =  £(a+8)  to  find  x. 

Ans.  x  =13. 


8.]  division.  75 

2.  Given  ^i-  +2x  =  ^=^  +16,  to  find  x. 

5  o 

an-         i«     .   «       O+14)(36x+10) 

3.  Given  16x  +  5  — Q   \  Q1   » to  finc*  x- 

JX— f—oi. 

Ans.  x  =  5. 

4.  Given  («c+fo-)2+52x:=  2«6er-f-(a2+£2)c2,  to  find 

-4ws.  x  =  e2 — r2. 

6.  Given  cc2-f-x~2  =  (x— x"1)  2-fx,  to  find  x  (§  49). 

Ans.  x  =  2. 

7.  A's  age  is  to  B's  as  2  to  3 ;  and  if  they  live  15  years, 
A's  age  will  be  |  of  B's.     What  are  their  ages  ? 

Let  x  =  B's  age ; 

then  %x  =  A's  age. 

Moreover  x-f-15,  and  §x-}-15  will  be  their  ages  after  15 
vears. 
•••  fx+l5  =  |(x+15). 

Ans.  A's  age,  30 ;  B's,  45. 

8.  A's  age  is  \  of  B's ;  and  18  years  ago,  A's  age  was 
B's.     What  are  their  ages  ? 


H.  DIVISION. 

§  78.  Division  is  the  process,  by  which,  having  a 
product  and  one  of  its  factors,  ive  find  the  other  fac- 
tor (see  §  10) ;  in  other  words,  it  is  the  process  of 
finding  hoiv  many  times,  or  parts  of  a  time,  one  quan- 
tity is  contained  in  another. 

Thus,  if  12  be  a  product,  and  3  be  one  of  its  factors,  the 
other  factor  is  4;   or  3  is  contained  in  12,  4  times;   if  12 
be  a  product,  and  24  be  one  of  the  factors,  the  other  factor 
•  or  24  is  contained  in  12,  \  a  time.  ' 


76  division.  [§79,80. 

DIVISION  OF  MONOMIALS. 

§  79.  As  in  multiplication,  so  in  division,  whatever  be 
the  quantities  involved,  the  operation  is  actually  performed 
upon  monomials  only.  "We  shall,  therefore,  consider  first 
the  division  of  monomials. 

Numerical  coefficients  are,  of  course,  subject  to  the  prin- 
ciples of  Arithmetic,  and  must  be  divided  accordingly.  Let- 
ters, we  have  seen,  are  divided  by  suppressing  in  the  divi- 
dend the  letters  of  the  divisor  (§  10.  b)  ;  i.  e.  by  subtracting 
the  exponents  of  the  letters  in  the  divisor  from  the  expon- 
ents of  the  same  letters  in  the  dividend  (§§16,  24.  b). 
See  also  §  13.  Hence,  we  have,  for  the  division  of  mono-* 
mials,  the  following 

RULE. 

§  80.  Divide  numerical  coefficients  as  in  Arithme- 
tic ;  and  annex  all  the  literal  factors,  which  remain 
after  suppressing  in  the  dividend  those  of  the  divisor, 

a.)  If  the  exponent  of  any  letter  be  greater  in  the  divi- 
dend, than  in  the  divisor,  its  exponent  in  the  quotient  will 
be  positive  ; 

If  equal,  it  will  be  zero  ;  i.  e.  the  letter  will  disappear ; 
and 

If  less,  it  will  be  negative. 

Or,  in  the  last  case,  the  division  may  be  expressed,  as 
we  have  seen,  by  placing  the  letters,  with  positive  expon- 
ents, as  the  denominator  of  a  fraction,  of  which  the  remain- 
ing factors  of  the  dividend  constitute  the  numerator  (§  10.  c). 

b.)  In  case  of  a  single  division,  we  have  shown,  that,  as 
in  multiplication,  like  signs  give  -\-,  unlike,  — .  In  case  of 
successive  division  by  several  divisors,  the  same  rule,  of 
course,  applies  to  each  operation.  Or,  bringing  the  signs 
together,  as  in  subtraction  (§  63)  and  multiplication  (§  68.  a) 
we  may  regard  only  the  negative  signs.  If  the  number  ot 
these  be  even,  the  quotient  is  positive  ;  if  odd,  negative. 


$81.]  MONOMIALS  AND  POLYNOMIALS.  77 

c.)  The  law  of  the  signs  may  be  otherwise  demonstrated, 
as  follows.  To  divide  by  any  quantity  is  the  same  as  to 
multiply  by  its  reciprocal  (§  19.  Cor.  IV.)  ;  and  the  recip- 
rocal of  a  quantity  evidently  has  the  same  sign  as  the  quan- 
tity itself  (§  18).  Therefore,  to  divide  by  any  number  of 
divisors  is  the  same  as  to  multiply  by  the  same  number  of 
multipliers  having  each  the  same  sign.  Hence,  the  law  of 
the  signs  is  the  same  in  division  as  in  multiplication. 

d.)  One  quantity  is  commonly  said  to  be  divisive  by 
another,  when  the  division  does  not  give  rise  to  fractional 
coefficients,  or  to  negative  exponents. 

Note.  Any  quantity  may  be  said  to  be  divisible  by  any  other. 
For,  whatever  be  the  dividend  and  given  factor,  another  factor  can 
always  be  found,  which  will  produce  the  dividend.  It  is,  however, 
convenient,  in  many  cases,  to  distinguish  as  perfect  or  exact,  the  di- 
visibility above  mentioned  which  does  not  give  rise  to  fractional  ex- 
preisions. 

1.  20a6b3c  —-  4a353c3  =  what  ?  Ans.  5a2c~2. 

2.  a2&~2-i-a-15  =  what?  a^x^-^-c^x-^y  ?  —  a^~-a~^? 

3.  amb-m-~a"b-n  =  what  ?     (ar\-x)  %-±(a+x)~%  ? 


TO  DIVIDE  A  POLYNOMIAL  BY  A  MONOMIAL. 

§  81.  In  multiplying  a  polynomial  by  a  monomial,  we 
multiply  each  term  of  the  polynomial  by  the  monomial,  and 
add  the  products  (§  69).  Therefore,  reversing  the  process, 
we  have,  for  dividing  a  polynomial  by  a  monomial,  the  fol- 
lowing 

RULE. 

Divide  each  term  of  the  dividend  by  the  divisor, 

and  add  the  quotients. 

Thus,  (ab  ±  ay)  ~  a  =  b  ±  y. 

1.   Divide  r?y-\-xy2  by  xy.  Quotient,  x-\-y. 

#7 


78  division.                              [§82. 

2.  {ax±x2)-:rx  =  what?     {rs—s)~s? 

3.  (2rx—x2)~x=\rhat?     {A2B2—B2x2)-±B2J 

4.  (— R  cos  b  cos  c  -\-  R  sin  b  sin  c)  -J i?  =  what  ? 

5.  (a— ;c) -^  a  z=  what  ?  Ans.  1 — a-1^  or  1- 

6.  Divide  Rz—x*  by  R*. 

7.  «     R  —  \R-^x2  —  \R-*x*  by  R. 

8.  "      a_t  —  §a~^ca  +  f  cT^ar*  by 


x 


a 


a  3. 


TO  DIVIDE  ONE  POLYNOMIAL  BY  ANOTHER. 

§  82.    Divide  3a&2-f3«2H-a3-H3  by  a+5. 

a.)  This  dividend  beiDg  regarded  as  the  product  of  the 
divisor  aud  quotient  (§  10),  the  terms  containing  the  highest 
and  the  lowest  powers  of  a  and  b  must  consist  of  the  unre- 
duced products  of  the  highest  and  of  the  lowest  powers  of 
those  letters  in  the  two  factors  (§  73.  1,  2). 

b.)  If,  therefore,  we  divide  the  term  of  the-  dividend 
which  contains  the  highest  power  of  a,  by  the  term  of  the 
divisor  which  contains  the  highest  power  of  the  same  let- 
ter, we  must  obtain  the  corresponding  term  of  the  quotient. 

c.)  If,  now,  we  multiply  the  divisor  by  the  term  of  the 
quotient,  which  we  have  found,  we  shall  have  one  of  the 
partial  products  whose  sum  is  the  dividend. 

d.)  If,  then,  we  subtract  this  partial  product,  there  will 
remain  the  sum  of  the  other  partial  products,  viz.  of  the  di- 
visor into  the  other  terms  of  the  quotient. 

e.)  There  will,  of  course,  be  a  highest  power  of  a  in  this 
new  or  remaining  dividend,  which  term  divided  by  the  term 
containing  the  highest  power  of  a  in  the  divisor,  as  before, 
will  give  a  term  containing  the  highest  power  of  a  in  the 
remaining  terms  of  the  quotient ;  and  so  on. 

/.  And,  as  the  sum  of  the  products  of  all  the  terms  of 
the  divisor  by  each  term  of  the  quotient  must  make  up  the 


§  82. J  POLYNOMIALS. 

dividend,  if  we  subtract  those  partial  products,  one  after1 
another  from  the  dividend,  they  must  exhaust  it ;  and  the 
remainder,  after  the  last  subtraction,  will  be  zero. 

g.)  If  we  obtain  a  remainder  equal  to  zero  by  simply  di- 
viding the  first  term  of  each  remainder  by  the  first  term  of 
the  divisor,  the  division  is  said  to  be  exact,  and  the  dividend 
is  said  to  be  divisible  by  the  divisor  (§  80.  d,  N.). 

h.)  If,  however,  after  exhausting  the  given  terms  of  the 
dividend,  wo  still  have  a  remainder,  the  division  may  be 
immediately  completed  by  writing  the  whole  remainder 
over  the  whole  divisor,  for  the  last  term  of  the  quotient ; 
or  the  division  may  be  still  farther  continued  (§  87)  accord- 
ing to  the  rule,  and  terminated,  whenever  we  please,  by  a 
fractional  term,  as  above  indicated. 

■i.)  These  operations  will  be  more  conveniently  'perform- 
ed, if  the  dividend  and  divisor  be  first  arranged  with  res- 
pect to  the  powers  of  some  one  letter  (§  33.  a). 

This  arrangement  may  be  according  to  either  the  ascend- 
ing or  the  descending  powers  of  the  letter.  The  descend- 
ing order,  however,  is  most  commonly  employed. 

k.)  The  polynomials  above  being  arranged  with  refer- 
ence to  a,  and  placed  in  order  for  division,  will  stand  thus : 
the  divisor  being  placed  at  the  right  of  the  dividend,  and 
the  quotient  under  the  divisor. 


a3-\-3a2b-\-3ab2-\-b* 

a+b 

a3-\-  a-b 

a2-\-2ab+b* 

2a2H-3a&2+£3 

2a25+2«52 

ab2+bs 

ab2+bs 

0 

From  the  reasoning  above,  we  deduce  the  following  gen^ 
eral 


DIVISION 


[8  88. 


RULE. 

§  83.  1.  Arrange  both  dividend  and  divisor  accord- 
ing to  the  powers  of  some  common  letter,  cither  as- 
cending, or  descending  in  both. 

2.  Divide  the  first  term  of  the  dividend  by  the  first 
term  of  the  divisor  (§  80),  and  set  the  result,  with  its 
proper  sign,  as  a  term  of  the  quotient. 

3.  Multiply  the  divisor  by  this  first  term  of  the  quo- 
tient, and  subtract  the  product  from  the  dividend. 

4.  Divide  the  first  term  of  the  remainder  by  the  first 
term  of  the  divisor,  set  the  result  in  the  quotient  with 
its  proper  sign,  multiply,  and  subtract  as  before,  and 
continue  the  process  as  long  as  the  case  may  require. 

1.  Divide  a*-\-3a2x+x3+3ax2  by  a+x. 

2.  Divide  x^Qy2x^ix^i-4:Xys-\-t/t  by  x*-\-2xy-\~yn-. 
a.)   It  is  not  necessary  to  write  all  the  remaining  terms 

of  the  dividend,  after  each  subtraction.  Indeed,  none  need 
be  written,  except  those  which  change  their  form  by  sub- 
traction and  reduction.  It  is  convenient,  however,  to  bring 
down  one  additional  term  of  the  dividend,  at  each  subtrac- 
tion.    This  is  the  method  commonly  practised. 

1.  Divide  a6— Sa^x-{-16a^x-— 20a3x3-\-15a*x*— Sax5 
-\-xz  by  a — x. 

2.  Divide  a±-\-a*zz+ z*  by  a*-\-az-\-z*. 

3.  Divide  x5-\-Sx*— 10x3— 112oc2— 207x— 110  by  ar2 
+7x4-10.  Quot.  x3— x2— 133—11. 

4.  Divide  aG—  3a4xz-\-3a2x'i— x*  by  a2—  x~. 

5.  Divide  1— a~n  by  a?— J-.  Quot.  a"^-f  a~i$. 

6.  Divide  ^a3— \a*b— ^ab2-{-^b3  by  \a-\b. 

Quot.  la'—lb"1. 

7.  Divide  x3-\-ax2 — bx*-\-cz2 — abx-\-acx— box— abc  by 
x  --\-ax — bx — ah. 


84.] 


Is*  Rem. 


POL 

^NOMIALS. 

x3-\-a 

x2— ah 

x — ahc 

x2-\-a 

x — ah 

—h 

-\-ac 
—he 

—h 

+c 

x-\-c  ( 

Quotient 

x3-\-a 

x2 — ahx 

—h 

cx2-\-ac 

x — ahc 

—he 

cx2-\-ac 

x — ahc 

—he 

81 


b.  The  operation  may  be  still  further  shortened.  Ar- 
range, divide  and  multiply,  as  directed ;  but,  instead  of 
writing  the  product  under  the  dividend,  subtract  each  term 
mentally,  as  it  is  formed,  and  write  the  reduced  remainder 
(§  83.  a).     Thus, 


a4— 4«3x+6a2x3— 4:ax3-\-x* 


1st  Rem. 
2d  Rem. 


-2a3x-\-5a2x2 — iax3 


a2—2ax-\-x2 


a2 — 2ax-\-x' 


a2x2 — 2ax3-\-x4: 


1.  Divide  x2 — 7x-\- 12  by  x — 3. 

2.  Divide  2a2'"-\-2a'"hp—  4a'V— 3amb— 3hp+*-\-§hcn  by 
2am—3b.  Quot.  am-\-hi>—2cn. 

§  84.  c.)  We  need  only  the  first  term  of  each  remainder 
(§  83.  4).  The  other  terms  are  simply  reserved  till  we  sub- 
tract from  them  the  terms  of  the  next  product,  and  so  on. 
Instead,  therefore,  of  performing  these  successive  subtrac- 
tions, we  may  write  the  similar  terms  of  the  several  pro- 
ducts under  one  another,  and  subtract  the  aggregate  of  each 
set,  when  the  corresponding  first  term  of  a  remainder  is  re- 
quired for  division. 

Or  we  may  change  the  signs  of  the  several  terms  of  the 
products  as  we  write  them,  and  add  each  column  as  we 
come  to  it.  If  we  adopt  this  course,  we  shall  be  less  liable 
to  mistake,  if  we  change  the  signs  of  the  divisor,  all  except 
the  first,  which  should  remain  unchanged,  to  prevent  mis- 
takes in  the  signs  of  the  quotient ;  and  which  can  occasion 


82  division.  [§  85. 

no  mistake  in  subtracting,  as  its  product  always  cancels  the 
term  above  it,  and  need  not  be  written. 

Divide  a*— 4a3a:+6a2a:2— 4aa;3-jr-a;4  by  a2—  2ax-\-x2. 


a4— ±a3x+Qa2x2— 4ax3-f-.T4 
-\-2g3x— a2x2 
— 2a3  x 

—±a2x2+2axs 


a2-\-2ax — cc2 


a2—2ax-\-x2 


-\-a2x7 

-\-2ax3 — x* 

d.)  The  last  method  is  conveniently  written  as  follows. 
Write  the  terms  of  the  divisor  under  one  another,  on  the 
left  of  the  dividend,  changing  the  signs  of  all  but  the  first. 
Write  the  terms  of  the  partial  products,  except  the  first  of 
each,  diagonally  under  the  corresponding  terms  of  the  divi- 
dend. Below,  in  a  horizontal  line,  write  the  first  terms  of 
the  remainders  as  they  are  formed,  each  under  the  column 
from  which  it  is  produced.  Write  the  quotient  also  in  a 
horizontal  line  below  the  last,  each  term  under  the  term  of 
the  dividend,  from  which  it  was  formed.  Thus, 
a4 — ia3x-\-6a2x2 — iax3-\-xi 
\-2ax      -\-2a3x—Aa2x2-\-2ax3 

—a2x2+2ax3—x* 


— x2 


— 2a3x-\-aux' 


Quotient,  a~ — 2  ax  -j-x2. 

Notes.  (1.)  If  any  term  in  the  series  of  powers  be  wanting,  its 
place  should  be  filled  with  a  cypher  (§76.  a);  or  the  given  terms 
should  be  placed  at  such  distances  from  each  other,  that  like  terms 
of  the  partial  products  may  stand  under  them.  (2.)  Each  term  of 
the  partial  products  will  stand  against  that  term  of  the  divisor  from 
which  it  is  formed. 

1.  Divide  a6-\-2a3z3-\-z6  by  a2 — az-f-z2. 

2.  Divide  a3-\-a2b — ah2 — b3  by  a — b. 

DIVISION  BY  DETACHED  COEFFICIENTS. 

§  85.  Division,  as  well  as  multiplication,  may  be  perform- 
ed by  DETACHED  COEFFICIENTS.      Thus, 


§  86.]  SYNTHETIC  DIVISION.  83 

1.    Divide  a3— 3a2£+3a&2+&3  by  a—b. 


1—3+3  -1 
1—1 


1—1 


1—2+1 


—2 
—2+2 


1 
1—1 


Supplying  the  letters,  by  dividing  the  first  term  of  the 
dividend  by  the  first  term  of  the  divisor,  we  have  a3 — 2c 
+&*. 

2.    Divide  a4—  6*  by  a2— 6s. 


1+0+0+0—1 
1+0—1 

0+1 

1+0—1 


1+0—1 


1+0+1.  .-.  Quot.=za*+b* 


SYNTHETIC  DIVISION. 

86.  Synthetic2  division  is  division  with  detached  co- 
efficients, performed  by  the  method  of  §  84.  cL  With  de- 
tached coefficients,  however,  the  method  admits  of  simplifi- 
cation, when  the  first  coefficient  of  the  divisor  is  1.  For, 
in  this  case,  the  coefficient  of  each  term  in  the  quotient  will 
be  the  same  as  the  corresponding  coefficient  of  the  first  term 
of  the  dividend  or  remainder ;  and  may,  therefore  be  found 
by  simply  adding  the  coefficients  above  it.  Thus,  to  divide 
a*— 4a3a+6«2^2— 4aa:3+a:4  by  a2— 2ax+«2. 


1 

+2 
— 1 


1—4+6—4+1 
+2—4+2 
—1+2—1 


1—2+1+0+0  .-.  Quot.  =  aB— 2ax-\-x-. 
Moreover,  if  the  fir3t  coefficient  of  the  divisor  be  not  1, 
it  can  evidently  be  made  so,  by  dividing  both  divisor  and 
dividend  by  the  given  first  coefficient. 

(x)  Gr.  cvv&ecic  ,  composition,  putting  together;  each  term  of 
the  quotient  being  formed  by  adding  the  like  terms  of  the  dividend 
and  of  the  pan  'acts  with  their  signs  changed. 


84  division.  [§  87. 

1.  Divide  x3— 3x2+3;r— 1  by  x2— 2a+l. 

2.  Divide  4a4— 9«252+6a63— 5*  by  2«2— 3a&+52. 

Solve  the  examples  of  §§  83,  84  by  this  method,  observ- 
ing, when  the  series  of  powers  is  not  complete,  to  fill  the 
place  of  the  missing  terms  with  cyphers  (76.  a). 

INFINITE  SERIES. 

§  87.  When  an  exact  division  is  impossible,  the  opera- 
tion may  still  be  carried  to  any  extent,  forming  what  is 
called  an  infinitev  series.  The  process  is  similar  to  the  pro- 
cess of  approximation  in  the  division  of  decimals  in  arith- 
metic. 

Thus,  to  divide  a  by  a-\-x. 


a 

a-\-x 


a-\-x 


1 — arlx-\-ar2x2 — a~3x3-\-a~4:x4: — &c. 


— x 

— x — a-1  a?2 
a   lx- 


a~1x2-\-a~2x3 


-a~2x3 

-a~2x3 — a~3x- 
a~3x* 


Or  (§  86),  thus, 


1 
— 1 


1 


-1+1— 1+1— &c. 


1—1+1— 1+1— &c.  .-.  Quot.  =  l— a-1x-^-a~2x2— &c 
"We  have,  therefore, 


a 


=  1— a-*x-\-a-2x2—  ar3x3-\-a-±x*— a"6a:B+&c.(l.) 
a-j-x 

or,  in  another  form  (§  14), 

ff  /y  T1-*  >y»3  t»4  <y5  nr*(j  n/*7 

a-\-x  a*   a2       a3   '   a4       a5'   a6       a7    '  v 

a.)  We  find  here  a  series  of  terms,  alternately  positive 
and  negative,  beginning  with  unity  or  the  zero  power  of 

(y)  Lat.  infinitus,  without  end* 


§  87.]  INFINITE  SERIES.  85 

both  x  and  a,  and  containing  in  the  successive  terms"  the 
powers  of  x  increasing,  and  those  of  a  decreasing  by  unity, 
without  limit,  that  is,  infinitely ;  the  numerical  coefficient 
of  each  term  being  unity.  As  soon  as  we  have  discovered 
this  order,  which  is  called  the  law  of  the  series,  we  may 
write  the  terms  to  any  extent,  without  the  labor  of  dividing. 
b.)  1.  Let  a=  10  and  x  =  1 ;  then 

10    _10       _  _1_ 1_      _L 1 

11  iaTiaa         -,  Ann  "PTaTvaTT       <£C. 


a+x      10+1"    11  10  ~  100      1000  "^  10000 

=  *  +  Too  +  loooo  +  &c'-(lo  +  1000+  &C')  = 

l.OlOlOl&c— (.1010101&C.)  =  .9090909&C. 

2.  Let  a  =  l,  x=10  ;  then 

-?—  =  TT  =  1  — 10+ 100— 1000+10,000 —  ^M^? — 
a-f-x        11  11       — 

(1  +  100  +  10,000)  -  (10  +  1000  +  1Q0'00-  )    - 
10,101  — 10,1001£  =  — • 

3.  Let  a  =  100  and  a;  =  l;  then -?-f  =  ^W  what  p 

4.  Let  a  =  1000,  and  x  =  1 ;  then  — ^—  =  what 9 

a+cc 

c.)  1.  Develop  (a+xWf— _2_ ). 

^     a+x  / 

^.  I-4+^!_&c.  or  I(i_£+!!!_^ 

a      a-      a3  «V       «~a2  7 

Let  a?=zl,  a  =  10,  100,  &c 

2.    Develop  (a— ar)-i.  ^w*.        i(l+-  +  — -kfec  \ 
Substitute  for  a  and  a?  as  above. 

5 


3.  Develop  6(a— or)-i  (  — -1— Y 


86  MULTIPLICATION  AND  DIVISION.  [$  88. 

4.  Develop  -—. — -.  Ans.  1 — u24-u* — z£6+&c. 

\-\-u'2  ' 

Letw  =  £,  i  TV,  &c. 

5.  Develop  H^H  =  frj^=gt+8i^0- 

^n5.—  (1 3-+&c). 

a-\         a        a2         a3  J 

d.)  Series,  in  which  the  terms  become  less  and  less  as  we 
proceed,  as  hi  b.  1,  3,  and  4,  above,  are  called  converging 
series,  and  are  of  great  utility  in  the  higher  applications  of 
Algebra.  When  the  terms  continually  increase,  as  in  b.  2, 
above,  the  series  is  called  diverging. 

Converging  series  may  be  treated  precisely  as  approxi- 
mating decimals  in  Arithmetic  ;  viz.  a  few  terms  may  be  ta- 
ken for  the  whole  series,  the  remaining  terms  being  so 
small,  that  they  may  be  neglected  without  sensible  error. 
Thus,  in  reducing  \  to  a  decimal  by  the  common  process, 
we  obtain  \^=.  .33333333  &c.  But,  we  may  apply  the  for- 
mula in  c.  3,  above,  by  making  b  =  3,  a—  10,  and  x  —  1 ; 

then =  — — -  =z  -  =  -.     Making  the  substitution,  we 

shall  have, 

e.)  In  c.  1,  2,  3,  and  5,  the  series  will  converge,  when- 
ever x<a ;  when  x  is  >«,  they  will  diverge.  The  series  of 
c.  4;  will  converge,  when  ^(<l. 

f.)  In  a  converging  series,  the  remainder,  after  a  few 
terms, may  be  neglected;  in  a  diverging  series, the  remain- 
der must  always  be  taken  into  the  account,  and  constitutes 
a  most  important  part  of  the  result. 

THEOREMS. 

§  88.  Algebra  employs  general  symbols  of  quantity  (§  1). 
Its  results,  therefore,  are  general  (§§  7.  a,  b,  55,  N.,  57.  3,  60. 


§89,90.]  theorems. (a  ±  b)^.  87 

4,  and  65.  N.)  ;  and  whatever  is  proved  of  numbers  repre- 
sented by  algebraic  symbols,  is,  of  course,  demonstrated*  of 
all  numbers  whatever.  General  truths  or  principles  thus 
demonstrated,  are  called  theorems". 

§  89.  Thus,  if  we  square  a-\-b,  we  have 
(a+J)  2  =  (a+b)  (a+b)  =  a*+2ab+b*.     That  is, 

Theorem  I.     The  square  of  the  sum  of  two  numbers  is 
equal  to  the  square  of  the  first,  plus  twice  the  product  of  the 
first  by  the  second,  plus  the  square  of  the  second. 

Or,  more  briefly, 

The  square  of  the  sum  of  txco  numbers  is  equal  to  the 
sum  of  their  squares,  plus  twice  their  product.  See  Geom. 
§  180.  Cor.  v. 

1.  (a+a:)2  =  what?  (a*-f-x3)2?  (a+2«)2?  (a*+&*)9? 

2.  (a5+&c)2r=what?     (x+J-p)2?     (l+2»m)2? 

So  in  Arithmetic ;  (16)2  =  (10+6)  »  =  102+2.10.6+62 
:=  100+120+36  =  256. 

(75)  2=  (70+5)  2  =  what?  (93) 2?  (II)2?  (19)9? 
73  =  (4+3)2?     (112)  2  =  (100+12) 2? 

Note.  An  absolute  equation  which  expresses  a  general  result 
or  a  theorem,  is  called  a  formula2. 

§  90.  If  we  put  — b  for  Vand  apply  the  principle  of  §  89, 
we  have, 

(a+(— &))  2—  (a_J)2—  a2_f-2a(— &)+(— b)  2  —a^—2ab 
+S2  [§11.  N.  2.].     Hence, 

The  or.  II.  The  square  of  the  difference  of  two  num- 
bers is  equcd  to  the  sum  of  their  squares,  minus  twice  their 
product.     See  Geom.  §  183.  Cor.  vii. 

Multiply  a — b  by  a — b,  and  see  if  the  same  formula  re- 
sults. 

1.  (x— ^)a=what?  (x— O2?  (y«—  y6)2?  See 
§  23.  d.     (x'  sin  a—i/sm  a')  2  ?  See  §  92.  N. 

(x)  Lat.  de:i:onstro,  to  show,  prove  beyond  the  possibility  of  doubt, 
(y)    Gr.  tieuprifia,  from  deupsu,  to  view,  contemplate.     (2)    Lat., 
form,  model,  rule. 


88 


SfULTIPLICATION  AND  DIVISION.        [§  91,  92. 


2-    (— ^+a;)2  =  what?     (2a2— 6a;2)2?     (1— <tM)2? 
Note.     We  have  evidently  (a— b)?  =  (b—a)2.     So  (10—1)2 
—  (i_io)2;  or  92  =  (—9)2     See  §11.  N.  2. 

In  like  manner  in  Arithmetic;  92:=(10  — 1)2  = 
102  —  2.10.1  + 12  —  100  —  20  + 1  =  81. 

(98) 2  =  (100  —  2)  2  =  what  ?  (75 2)  =  (80  —  5) 2  ? 
(47)2=  (50  —  3)2?     (93)2  =  (100  — 7)2=  (90  +  8)2? 

§91.   a.)  (a-\-b)2  =  a*-\-2ab+b2.  §89. 

And  (a—b)  2  =  a2— 2ab-\-b*.  §  90> 

.-.   Adding  and  subtracting  the  equations 
(a-f-Z>)2+(«— *>y  =  2a2+252  =  2(a2+52).    Geom.  §  199. 
(a-f-J)  ^— (a—b)  2  =  Aab.     Geom.  §  184.     Hence, 

Cor.  (1.)  The  square  of  the  sum,  plus  the  square  of  the 
difference,  of  two  numbers,  is  equal  to  twice  the  sum  of  their 
squares.  (2.)  The  square  of  the  sum,  minus  the  square  of 
the  difference,  of  tivo  numbers,  is  equal  to  four  times  their 
product. 

§  92.  Multiply  a~\-b  by  a—b. 
We  have,  («+&)  («— &)  =  a  2 — b  2 .     Hence, 

Theor.  III.  The  product  of  the  sum  and  difference  of 
two  numbers  is  equal  to  the  difference  of  their  squares. 
See  Geom.  §  185.  Cor.  ix. 

1-    (H-f)(*-f)  =  what?     (A+x)(A-x)?     (y-h/O 

(2/-y) ?   ((i+x)h+(i-x)*)«i+xy-(i-x)h 

2.  (R+x) (R—x)  =  what  ?  (AB+BC) (AB—BG)  ? 
(*2-h/2)(*2-2/2)  ?      (a!8_|^8).(a.a_Sfa)  ? 

3.  (sin  a  cos  &-(-sin  5  cos  a)  (sin  a  cos  b — sin  b  cos  a)  — 
what?  Ans.  sin2«cos2£> — sin2£cos2a. 

Note  .  Sin2a  denotes  the  square  of  the  sine  of  a.  This  notation 
is  more  precise  than  sin  a%,  which  might  mean  the  sine  of  the  square 
of  a;  and  is  less  cumbrous  than  (sin  a)-.  The  same  remark  applies 
to  cos'a,  tan2a,  &c 


4  93,  94]    THEOREMS.  —  SUM  AND  DIFFERENCE.  89 

4.  (a+J+c)  (a-\-b—c)  [i.  e/the  sum  of  a-\-b  and  c,  into 
the  difference  of  a-\-b  and  c]  =  -what  ? 

^ws.  (a+b)  2— c2  =  a2+2a5+52— c2. 

5.  (a+b—c)  (a—b+c)  (  =  (a+5— c)(a— J— c))  =  what  ? 
So  in  Arithmetic ;  12X8  =  (10+2) (10-2)  =  102— 22 

=  100—4  =  96. 

19X21(  =  (20— 1) (20+1))  =  what?  103x97?  51X49? 
101X99?     1004X996?     1000£X999£? 

§  93.  The  same  formulas,  read  with  the  second  member 
first,  give  the  converse"  of  the  above  theorems ;  and  enable 
us  to  resolve  several  classes  of  polynomials  into  their  fac- 
tors (§  75).     Thus, 

(I.)    The  sum  of  the  squares,  plus  twice  the  product,  of 
two  numbers,  is  equal  to  the  square  of  their  sum. 

1.  aj2+2a;?/+3r  =  what?  Ans.  {x-\-y)2. 

2.  2/2+2yy/+y/2  =  what?  l+2?i+?i2  ?  9a4+24a262 
+1664  ?     169  (=  100+2.10.3+9  =  102+2.10.3+32)  ? 

(II.)    The  sum  of  the  squares,  minus  twice  the  product,  of 
two  numbers,  is  equal  to  the  square  of  their  difference. 

1.  x2— pa+£p2  =  what?  Ans.  {x— ^p)2. 

2.  b-—  2Z>c+c2  =  what  ?  1— 4ra+4n2  ?  a2—  12ab 
+3662?     81(  =  100— 2.10.1+1  =102— 2.10.1+1 2)? 

(III.)  The  difference  of  two  squares  is  equal  to  the  pro- 
duct of  the  sum  and  difference  of  their  roots  (§  23). 

1.  i?2— z2  =  what?  Ans.  (H-\-x)(E—x). 

2.  sin2  a— sin2  6  =  what?     x*—y*?      {AB)2—[BC)2? 

„G_&6  ?      c2_£2_|_2fo— c2(  —  a2_(J_C)2  [§§  63. 1,  90.])  ? 

£2_|_26c+c2— « 2  ?     l_cos2i<  =  i  2— cos2*)  ?     x2— x"2  ? 

§  94.   1.    Divide  a2— b2  by  a—b.  Quot.  a-\-b. 

(a)  Lat.  conversus,  turned  about.  Of  two  propositions  or  sen- 
tences, each  is  said  to  be  the  converse  of  the  other,  when  the  condi- 
tion of  the  first  is  the  conclusion  of  the  second,  and  the  conclusion 
of  the  first  is  the  condition  of  the  second;  or  when,  in  like  manner, 
subject  and  predicate  change  places.     See  Geom.  §32.  Note  n. 

*8 


90  division.  [§  95. 

2.  Divide  a3— b3  by  a— b.        Quot.  a*-\-ab-\-b2. 

3.  «      a4— 54  by  a— i.  "  a3+a26+a&2+63. 

4.  "      an— bn  by  a— 5. 
Employ  tbe  method  of  §  84.  d;  thus6, 

aan —b'c. 


+& 


Guo*.  =  a^1+a,,-3H-«,,~3i2+ -\-abn-2+bn-K 

Now,  in  the  successive  terms  of  the  remainders,  the  ex- 
ponents of  a  diminish,  and  those  of  b  increase  by  unity. 
The  terms  of  the  quotient,  of  course,  follow  the  same  law ; 
and  the  sum  of  the  exponents  in  each  term  of  the  quotient 
is  n — 1.  Hence  we  shall  find  a  term  containing  a0  and 
J*-1 .  This  term,  multiplied  by  b,  will  give  bn,  which  ad- 
ded, will  cancel  — bn  in  the  dividend,  and  leave  a  remain- 
der equal  to  zero,  showing  a  perfect  division  (§  82.  g). 

§  95.  a.)  Otherwise, 
an—  bn  a  —  b 


a—ib  —  bn=(an-1—  b"~l)b. 

Now,  if  a—b  will  divide  (§  82.  g)  this  remainder,  it  will, 
obviously,  divide  the  whole  dividend.  But  it  will,  evident- 
ly, divide  this  remainder,  if  it  will  divide  one  of  its  factors, 
a»-i_  fr'-i.  Hence,  if  an~l  —  b"~l  is  divisible  by  a  —  b, 
aa  _  ^  ig  ^  divisible  hy  a  —  b.  That  is,  if  the  difference 
of  like  powers  is  divisible  by  the  difference  of  their  roots, 
the  difference  of  the  powers  of  the  next  higher  degree  is 
divisible  in  like  manner. 

But  we  have  found  a4— &4  divisible  by  a—b;  .:  a^—b5 
is  divisible  by  a—b;  so  ae—be,  a"—b7  ;  and  so  on,  with- 
out limit.  Or,  a1—  bl  is  divisible  by  a—b;  .:  a2—b2  is  di- 
visible by  a—b;  and  so  on.     Hence  (§§  94,  95), 

(b)  Here,  the  sign  of  b  being  changed,  the  second  term  of  each 
product  is,  without  any  reduction,  the  first  term  of  the  corresponding 
remainder,  and,  of  course,  need  be  written  but  once.  We  cannot 
write  all  the  terms  of  the  quotient,  unless  we  assign  a  particular  val- 
ue to  n.     The  whole  number  of  terms  is,  obviously,  equal  to  ?». 


§96-97.]  THEOREMS. — DIVISIBILITY.  91 

§  96.  Theor.  IV.  The  difference  of  any  two  posi- 
tive integral  powers  of  the  same  degree  is  divisible 
by  the  difference  of  their  roots. 

Notes.  (I.)  This  method  of  proof  is  of  great  utility  in  Algebra, 
and  should  be  perfectly  understood.  It  consists  in  showing,  that,  in 
how  many  soever  instances  a  principle  has  been  found  true,  it  will 
be  true  in  one  instance  more.  If  it  be  true  in  n — 1  cases,  it  will  be 
in  n;  if  in  n,  then  in  n+l ;  if  it  be  true  in  one  instance,  it  will  be 
true  in  the  second;  if  in  the  second,  then  in  the  third,  and  so  on. 
(2.)  The  limitation  to  "positive  integral  powers"  is  necessary; 
for  the  principle  has  been  applied  to  such  powers  only.  And,  if  n — 1 
is  a  positive  integer,  n,  obviously,  cannot  be  either  negative  or  frac- 
tional. 

a^-Jf  _  an_i  ^  a>i_2b  + _j_a5»-2  _}_  J«-i. 

a — b 

a» bn 

b.)  If  a  =  b,  we  have =  m"_1;    a   result   which 

a — a 

will  be  considered  hereafter. 

c.)    a_n_J-n_(a_i)»_(J-l)».  §2±.d. 

a~n  —  b~n  is  divisible  by  ar~ 1— [b—1.     §  96.     Or, 

which  is  the  same  thing, -—  —  7- is    divisible   by -7. 

0   an       b"  a      b 

d.)  ci^m  —  b^m  (  =  (o^)"-  (fi^m)")  is,  evidently,  di- 

±-  z±- 

visible  by  a  m  —  6  »<• 

§97.  e.)  a2"— i2"(  =  (o9)n— (52)")  is  divisible  by  a2— b- 
(§  96),  i.  e.  by  {a~\-b)(a— b). 

a-n  —  b2n  is  divisible  by  a +  b. 

Now  2m,  being  divisible  by  2,  is  an  even  number.    Hence, 

Cor.  1.    The  difference  of  any  two  even  positive  integral 
powers  is  divisible  by  the  sum  of  their  roots. 

To  divide  a-n—b-n  by  «+&,  employ  the  method  of  §  86. 

-1+1-1+ -1+1-1+1 


1 

IK 

—  1 


1_1_L.1_1+ _l_|_l_l_|-0 

(ai"—b2n)-±-{a+b)  =  a2"-1  —  a2"-2&  +  a2*-3Z>2—  .  .  . 

+  «3&2»-4  _  a2Z,2"-3  _L.  ab~n~~  —  b2n~l. 


"2  division.  [§  98-100- 

Thus,  if  2n  =  4, 

(a*  —  b±)  +  {a-{-b)  =  a3—a2b-\-a?>S  —  b3, 
*  98.  /.)   Divide  a2"+i  +  &2«+i  bya  +  b, 


a2«+l  _|_J2«+1 


a-f-^ 


a2"— &c. 

But  &2n—  a3"  is  divisible  by  a  +  5  (§  97). 

as«+i  _j_  J2H-1  is  divisible  by  a  +  b. 

Moreover  2w-f-l  is  an  odd  number,  being  greater  by  uni- 
ty than  2n,  an  even  number.     Hence, 

Cor.  ii.  The  sum  of  any  two  odd  positive  integral  pow- 
ers of  the  same  degree  is  divisible  by  the  sum  of  their  roots. 

a2>4-l_l_52>4-l 

i- =  «2n  —  a2n~ib  +  a2n~*b2  —  a2"-3£3  4- 

.        .       H-G462H-*  — a352"-3-f  «233»-2_aJ2;»-l_|_J2r.> 

Thus,  if  2n+l  =  5, 

§  99.  The  principles  of  §  94-98  obviously  enable  us  to 
resolve  another  large  class  (§  93)  of  polynomials  into  their 
factors.     Thus, 

a3+b3  =  (a-\-b)(a*+ab+b*). 
What  are  the  factors  of  a4— 64?    of  a5— br>?    of  a 5+3 5? 
of*3— 27(=x3— 33)?     of  aG—x*?     ofa:3+64? 

THE  GREATEST  COMMON   DIVISOR. 

$100.  A  factor  common  to  two  or  more  quanti- 
ties is  called  a  common  divisor ;  and  the  greatest 
common  factor,  i.  e.  the  product  of  all  the  common 
factors,  is  called  the  greatest  common  divisor,  or  mea- 
sure. 

Thus,  of  the  quantities  18abx  and  20a^by,  2,  a  and  b  are 
common  divisors ;  and  the  greatest  common  divisor  is  evi- 
dently 2ab,  the  product  of  all  the  common  factors. 


§  101,  102.]      GREATEST  COMMON  DIVISOR.  93 

Notes.  (I.)  The  term  divisor  is  used  here  with  reference  to 
perfect  divisibility  (§  82.  g).  (2.)  A  single  factor,  as  a,  which  has 
no  integral  divisor  but  itself  and  unity,  is  called,  as  in  Arithmetic,  a 
prime  factor.  (3.)  Quantities  which  have  no  common  divisor  but 
unity  are  called  incommensurable,  or  prime  to  each  other. 

a.)  One  of  two  or  more  quantities  may  be  either  multi- 
plied or  divided  by  any  factor  not  found  in  all  the  other 
quantities,  without  affecting  the  greatest  common  divisor. 
For  the  factor  so  introduced  or  taken  out,  not  being  com- 
mon to  all  the  quantities,  can  form  no  part  of  their  greatest 
common  divisor. 

Thus,  the  greatest  common  divisor  of  \'2ax  and  20ay  is 
the  same  as  that  of  I2ax  and  20ayX5b  or  20ay-:roy. 

§  101.  The  greatest  common  divisor  of  several  monomi- 
als must  evidently  consist  of  all  the  common  literal  factors, 
multiplied  by  the  greatest  common  divisor  of  the  numerical 
coefficients. 

What  is  the  greatest  common  divisor  of  12a3a:4  and 
2«2X5?  of  A-x"y"  and  B2x"y"?  of  ax  and  a'x?  of 
A  -cy"  and  c *xf'y"  ?     of  nxn~ 1 0+1)"  and  nxn(x+l)"- 1 . 

§  102.  The  process  of  finding  the  greatest  common  divi- 
sor of  two  polynomials  is  substantially  the  same  as  that 
employed  in  Arithmetic,  and  depends  on  the  following 
principle ;  viz. 

The  greatest  common  divisor  of  two  quantities  is 
the  same  as  the  greatest  common  divisor  of  either  of 
them,  and  of  the  remainder  obtained  by  dividing  one 
by  the  other. 

To  prove  this,  let  the  two  quantities  be  A  and  B,  and  di- 
vide A  by  B.  Let  the  integral  quotient  resulting  from  this 
division  be  Q,  and  the  remainder  R.  Then  A — BQ  —  R, 
ox  A  —  BQ+R. 

Now  every  divisor  of  Bis,  of  course,  a  divisor  of  BQ. 
Therefore  every  common  divisor  of  A  and  B  is  a  common 
divisor  of  A  and  BQ.     Also,  every'such  common  divisor  is 


<J4  division.  [§  103,  104. 

a  divisor  of  A—B  Q  [§  8 1  ]c,  i.  e.  of  R.  That  is,  every  com- 
mon divisor  of  A  and  B  is  a  common  divisor  of  B  and  i?. 

Again  every  common  divisor  of  B  and  R  will  divide  i?<> 
and  i?,  and,  of  course,  ^^+i?  or  A.  That  is,  every  com- 
mon divisor  of  A  and  B  is  a  common  divisor  of  B  and  i?. 

Hence,  the  greatest  common  divisor  of  B  and  R  is  the 
greatest  common  divisor  of  A  and  B. 

§  103.  By  the  same  reasoning,  if  we  proceed  to  divide  B 
by  R,  and  obtain  a  remainder  R',  the  Greatest  common  di- 
visor  of  R  and  i2'  is  the  greatest  common  divisor  of  B  and 
R,  and,  therefore,  of  A  and  i?. 

Thus,  the  greatest  common  divisor  of  any  of  these  divi- 
sors and  its  remainder  is  the  greatest  common  divisor  of  all 
the  preceding  remainders,  and  also  of  the  original  quanti- 
ties. If  then  we  find  a  remainder,  which  divides  the  pre- 
ceding remainder,  it  is  the  greatest  common  divisor  re- 
quired. 

a.)  If  the  first  term  of  any  dividend  be  not  divisible  by 
the  first  term  of  the  corresponding  divisor,  we  must  (1.) 
suppress  any  factor  of  the  divisor,  not  found  in  the  divi- 
dend ;  and  (2.)  we  may,  if  necessary,  multiply  the  dividend 
by  any  factor  not  found  in  the  divisor  (§  100.  a). 

Note.  If  we  suppress  in  the  divisor  a  factor  found  also  in  the 
dividend,  that  factor,  originally  common,  will  cease  to  be  so,  and  the 
common  divisor  will  be  less  than  it  ought  to  be.  If,  on  the  other 
hand,  we  introduce  into  the  dividend  a  factor  already  found  in  the 
divisor,  that  factor,  not  originally  common,  will  become  so,  and  the 
common  divisor  will  be  greater  than  it  ought  to  be  (§  100). 

Hence,  to  find  the  greatest  common  divisor  of  two  quan- 
tities, we  have  the  folio  wing 

RULE. 

§104.  Divide  one  quantity  by  the  other;  then  di- 
vide the  first  divisor  by  the  first  remainder^  the  second 

(c)  If,  for  instance  A-—D  and  BQ-hD  are  both  whole  numbers, 
their  difference  or  their  sum  must  be  a  whole  number. 


§  104]  GREATEST  COMMON  DIVISOR.  95 

divisor  by  the  second  remainder,  and  so  on ;  alwetys 
rendering  the  first  term  of  the  dividend  divisible  by  the 
first  term  of  the  divisor  (§  103.  a).  The  divisor  which 
gives  no  remainder,  is  the  greatest  common  divisor 
required. 

a.)  If  the  first  remainder  which  divides  the  preceding 
remainder  be  unity,  the  quantities  are  said  to  have  no  com- 
mon divisor,  but  to  be  incommensurable,  or  prime  to  each 
other. 

b.)  If  the  greatest  common  divisor  of  more  than  two 
quantities  be  required,  we  must  first  find  that  of  two  of 
them,  and  then  of  that  divisor  and  a  third,  and  so  on. 

1.    Find  the  greatest  common  divisor  of  98  and  112. 

112  98 

98  ~T 
7   .'.14  is  the  greatest  common 


98 

98 
divisor  required. 

Note.  In  Arithmetic,  it  is,  of  course,  proper  to  divide  the  great- 
er number  by  the  less.  In  Algebra,  the  quantity  containing  the  high- 
est power  of  the  letter  of  arrangement  will  be  the  first  dividend.  If 
the  highest  power  is  the  same  in  both,  either  may  be  made  the  divi- 
dend. 

2.   Find  the  greatest  common  divisor  of  cc2-f-5rc-{-6  and 
x*+2x— 3. 

x*+  5x-f-6  x2+2x— 3 


ar2+2x— 3 


i 


3x+9  =  3(aH-3).  Reject  3  (103.  a). 


x2-{-2x—3 
x*-\-3x 


x+3 


x—  1 


-x—  6 
-x—3 


x-{-3  is  the  greatest  common  divisor. 
3.    Find  the  greatest  common  divisor  of  a2-{-2ax-\-x- 
and  a3 — ax2.  Ans.  a-\-x. 


96  division.  [§  105,  106. 

4.  Find  the  greatest  common  divisor  of  9x3-\-53x2 — 9x 
—18  and  x2-\-l ls+SO.  Ans.  x+6. 

5.  Find  the  greatest  common  divisor  of  2x3-\-x2 — 8rtr-}-5 
and  7a;2 — \2x-\-5.  Ans.  x — 1. 

6.  Find  the  greatest  common  divisor  of  a3x-\-2a-x- 
-}-2ax3-}-^4  and  6a5-\-\0a*x-\-oa3x'2. 

§  105.  c.)  The  application  of  the  above  rule  (§  104)  to 
polynomials  is  simplified  in  various  ways.  Thus,  before 
applying  the  rule, 

1.  Any  factor  obviously  common,  may  be  taken  out,  and 
reserved,  as  a  factor  of  the  common  divisor  required  (§  100). 

2.  Any  factor,  found  in  a  part  only  of  the  polynomials, 
may  be  rejected  (§  100.  a). 

3.  If  one  of  two  polynomials  contain  a  letter  not  found  in 
the  other,  the  common  divisor,  obviously,  cannot  contain 
that  letter,  i.  e.  must  be  independent  of  it,  and  must  there- 
fore be  the  common  divisor  of  the  coefficients  of  the  seve- 
ral powers  of  that  letter.  In  this  case  it  is  often  best  to 
arrange  the  polynomials  with  reference  to  that  letter,  and 
to  find  the  greatest  common  divisor  of  its  coefficients. 

Note.  1  and  2,  above,  are  most  easily  applied  to  monomial  factors 
of  the  polynomials;  for  such  factors  can  always  be  found  by  inspec- 
tion (§§  69,  81).  But  they  are  equally  applicable  to  polynomial  fac- 
tors, when  we  can  discover  them  (§§  93,  99). 

1.  Find  the  greatest  common  divisor  of  a3-\-2a*x-{-ax- 
and  5ab3 — oabx-.  Ans.  a(a-\-x). 

2.  Find  the  greatest  common  divisor  of  x3-\-ax^-\-bx- 
— 2a-x-\-abx — 2«25  and  x'2-\-2ax — bsc — 2ab. 


COMMON  MULTIPLE. 

§106.  A  common  multiple  (§46.  a)  of  two  or  more 
quantities  is  a  quantity  which  each  of  them  will  divide 
(§  80.  d).  The  least  common  multiple  is  the  least  quantity 
which  they  will  divide. 


§  107-109.]  COMMON  MULTIPLE.  97 

§  107.  A  quantity  is  evidently  a  multiple  of  any  other 
quantity  of  which  it  contains  all  the  factors ;  if  it  contain 
the  factors  of  each  of  several  quantities,  it  is  their  common 
multiple ;  if  it  contain  each  factor  no  often er,  than  some 
one  of  the  quantities,  it  is  the  least  common  multiple.  That 
is, 

The  least  common  multiple  of  several  quantities  consists 
of  all  their  factors,  each  tvith  the  highest  exponent  which  it 
has  in  any  of  the  quantities. 

Thus,  the  least  common  multiple  of  Qab-(  =  2.3ab2)  and 
9a2c(=:32a2c)  is  2.32a262c  =  18a262c. 

§  108.  The  least  common  multiple  of  two  quantities  con- 
sists of  all  their  prime  factors,  each  with  its  greatest  expon- 
ent (§  107) ;  and  the  greatest  common  divisor  consists  of  the 
common  prime  factors,  each]with  its  least  exponent  (§  100). 
Therefore, 

The  least  common  multiple  of  two  quantities  is  equal  to 
their  product  divided  by  their  greatest  common  divisor. 

Thus,  the  greatest  common  divisor  of  x-y  and  xy2  is  xy ; 
their  product  is  x3y3  ;  and  their  least  common  multiple  is 
x2y2=zx3y  z-—xy. 

Note.  Every  algebraic  factor  of  the  first  degree,  whether  mono- 
;r,ial  or  polynomial,  is  a  prime  factor. 

PROBLEMS. 

X2 
§  109.    1.   Given  4a4-x  =  - — — ,  to  find  x. 

4a-[-x 

x            x 
2.    Given  — —  -j =  2,  to  find  x. 


a-\-b       a — b 

ax     ,      a 
: to 

a — x       a-\-x 

a2+a3— 1 


3.    Given  — —a  = 1 — ,  to  find  x. 

a2 — x*  a — x       a-\-x 


Ans.  x  = 

a — a'J 

4.   A  sum  of  money,  x,  is  divided  among  several  per- 
sons, so  that  A  receiving  $1000   lesg   than  half,  and  B, 
alg.  9 


98  MULTIPLICATION  AND  DIVISION.  [§  109. 

$1000  move  than  one  third,  of  the  -whole,  find  their  por- 
tions equal.     What  is  the  value  of  x  ? 

5.  Let  A  receive  a  less  than  half,  and  B,  a  more  than 
one  third,  and  let  their  portions  be  equal.  What  is  the 
value  of  x  ? 

G.  Two  couriers  are  traveling  on  the  same  route,  and  in 
the  same  direction.  A  is  100  miles  in  advance  of  B,  and 
travels  10  miles  an  hour,  while  B  follows  at  the  rate  of  1 2 
miles  an  hour.     In  how  many  hours  will  they  be  together  ? 

Let         x  =  the  number  of  hour.:. 

Then  10cc  =z  the  distance  A  will  travel , 

and         12x  =  the  distance  B  will  travel,  before  they  come 
together. 

Now,  if  B  overtake  A,  he  must  travel  as  many  miles  as 
A,  and  the  distance  between  them,  100  miles,  in  addition. 
12.r=10x+100;  or  12x—  10z=:100. 
x  ■=.  50  hours. 

7.  In  what  time  will  they  be  together,  if  A  goes  10  miles 
an  hour,  and  B  1 1  ? 

8.  In  what  time,  if  A  goes  10  miles  an  hour,  and  B  10^  ? 
A  10,  and  B  10^?     A  10,  and  B   10^?     A  10,  and  B 

10rio?     A10,andB10TIJW?     A  10,  and  B  lO^Voo  ? 

9.  In  what  time,  if  A  goes  10  miles  an  hour,  and  B  10  ? 
In  the  last  case,  we  have  lQx — 10x=z  (10 — 10)x=  100. 

_     100       _100 

x  ~  io=io  -"o~" 

a.)  How  shall  this  result  be  interpreted  ?  If  we  divide 
100  by  .01,  .001,  .0001,  .00001,  &c,  the  quotient  obviously 
increases  as  the  divisor  diminishes,  and  in  the  same  pro- 
portion. Consequently,  if  the  divisor  becomes  numerically 
less  than  any  quantity  whatever,  or  0,  the  quotient  must 
become  greater  than  any  quantity  whatever,  i.  e.  infinite. 
For  no  number  can  be  assigned  or  conceived,  so  great  as, 
when  multiplied  by  0,  to  produce  100.  Hence  i#£,  or,  in 
general,  ~-  (a  being   any    quantity  whatever,  numerically 


§  109."|  PROBLEMS. 


' 


greater  than  0),  is  infinite,  i.  e.  greater  than  any  assignable 
quantity;  and  is  expressed  by  the  symbol  oo. 

Now,  as  the  difference  of  the  rates,  in  the  preceding  problems,  be- 
came  less  (i.  e.  as  B  gained  less  in  an  hour),  the  number  of  hours 
required  for  him  to  overtake  A  became  greater.  When  the  difference 
of  tho  rates  is  nothing,  the  time  will  be  infinite  (i.  e.  B  will  never 
overtake  A).  In  other  words,  if  B  gains  nothing  in  one  hour,  no 
number  of  hours  can  enable  him  to  gain  100  miles. 

10.  Again,  suppose  that  A  travels  10,  and  B  8  miles  an 
hour,  when  will  they  be  together  ? 

Here  we  have  8x— 10*  =  —  2x— 100.  .-.  x~—  50  (§  5). 
That  is,  A  and  B  xvere  together  50  hours  ago. 

Note.    Had  it  been  proposed  to  find  when  they  had  been  togeth 
er,  the  answer  would  have  been  positive  (§4.  c). 

11.  Let  A  be  a  miles  in  advance  of  B  ;  and  let  A  trav- 
el n,  and  B  m  miles  an  hour.  When  will  they  be  togeth- 
er ?  Ans.  In hours. 

m — n 

b.)  The  last  problem  is  the  generalization  of  the  preced- 
ing (6-10).  We  shall  evidently  have,  if  «>0  (§  6.  a),  when 
?»>»,  m — n  positive,  and,  of  course,  the  result  positive  ; 
when  m  =  n,  m — n  —  0,  and  the  result,  infinite  ;  and  when 
}«<«,  m — n,  negative,  and  the  result  negative  (§  10.  d).  If 
a  ■=.  0,  and  m>,  or  <«,  the  result  is  0  (i.  e.  they  are  togeth- 
er now)  ;  and  if  m  =  n,  the  result  is  %  (i.  e.  they  are  togeth- 
er now,  and  must  always  remain  together  [§  109.  c]). 

12.  Let  them  travel  towards  each  other,  A,  n,  and  B,  m 
miles  an  hour.     When  will  they  meet? 

Ans.   In  — ; —  hours. 
m-yn 

Let  a  — 100,  m  =■  12,  and  n  =  8 ;  &c. 

Note.  The  formula  of  12,  above,  includes  this  case  also.  For 
the  rate  or  velocity  of  one,  being  positive,  and  represented  by  m, 
that  of  the  other  must  be  negative  (§5),  and  may  be  denoted  by — n; 
and  the  difference  of  the  rates  will  be  properly  expressed  by  m — ( — n) 
=  m  +  n.     Hence,  we  have  [12], 

a  a 

~  rn — ( — n)      m-\-ri 


100  MULTIPLICATION  AND  DIVISION.  [§  10^. 

14.  The  age  of  a  father  is  36  years ;  that  of  his  son  is 
12.  In  how  many  years  will  the  age  of  the  father  be  just 
double  that  of  the  son.  Ans.  12. 

15.  In  how  many  years  will  it  be  triple? 

Ans.  0  (i.  e.  it  is  triple  now). 

1 6.  In  how  years  will  it  be  quadruple  ?         Ans.  — 4. 
That  is,  it  tvas  quadruple  4  years  ago.     If  we  had  inquired,  hovr 

long  since  it  had  been  quadruple,  the  result  would  have  been  posi- 
tive. 

17.  In  how  many  years  will  the  ages  be  equal  ? 
Ans.  co  (i.  e.  they  will  never  be  equal  [§  109.  «]). 

18.  Let  A's  age  be  a,  and  B's,  b  years ;   in  how  many 

d — ?2  J 

years  will  A's  age  be  n  times  B's  ?  Ans. . 

J  n—\ 

Here,  if  «>1,  the  result  will  be  positive  (i.  e.  the  event  will  be 

future),  when  a^>nb;   negative  (i.  e.  the  event  will  be  past),  when 

a^nb;  and  zero  (i.  e.  the  event  will  be  present),  when  a  =  ?i6.     If 

n  =  1,  the  result  will  be  ±  oo,  when  a>  or  <^6 ;  and  the  result  will 

be  Sl,  when  a^=.b.    If  7i<l,  the  result  will  be  positive,  when  a^nb; 

negative,  when  a'p-nb;  and  zero,  when  a  =  nb. 

c.)  In  regard  te  the  result  -g-,  it  is  obvious,  that  any  finite 
quantity  whatever,  multiplied  by  the  divisor,  0,  will  pro- 
duce the  dividend,  0,  and  is  therefore  a  proper  value  of  the 
expression.  This  expression  may  therefore  represent  any 
quantity  whatever,  and  is  henco  called  an  indeterminate  ex- 
pression. 

Thus,  in  problem  12,  if  a  —  0,  and  m  =  n  (in  which  case 
the  result  becomes  g),  A  and  B  are  together  now,  and  must 
always  remain  together.  Hence,  any  number  of  hour- 
whatever  will  truly  express  the  time  at  the  end  of  which 
they  will  be  together.  We  may,  of  course,  have  an  infin- 
ite number  of  solutions ;  and,  as  no  one  of  these  is  better 
than  another,  the  problem  is  said  to  be  indeterminate. 


CHAPTER  III. 


FRACTIONS. 


§  110.  A  fraction,  in  Algebra  as  in  Arithmetic,  is  the 
expression  of  a  division  (§§  2./.  N.,  10.  c). 

Thus  |  and  j  express  the  division  of  3  by  4  and  of  a 
by  J. 

§  111.  Again  a  fraction  may  be  regarded  as  expressing 
equal  parts  of  a  unit;  the  denominator6  showing  the  na- 
ture of  the  parts,  and  the  numerator0,  the  number  of 
them  employed. 

Thus  |  shows,  that  the  unit  is  divided  into  4  equal  part*, 

and  that  3  of  them  are  taken.     So  T  shows,  that  the  unit 

b 

is  divided  into  b  equal  parts,  and  that  a  of  them  are  taken. 

Note.  The  numerator  and  denominator  are  called  the  terms  of  a 
fraction. 

§  112.  A  quantity  expressed  without  the  aid  of  fractions 
is  called  entire  or  integral.  An  expression  partly  entire 
and  partly  fractional  is  call :  d  mixed. 

§  113.  Operations  upon  fractions  are  of  the  same  nature 
in  Algebra  as  in  Arithmetic ;  and  depend  on  the  following 
principles,  which  we  shall  here  assume  without  demonstra- 
tion. 

1 .  If  the  numerator  of  a  fraction  be  midtiplied  or  divided, 
the  fraction  itself  is  equally  midiiplied  or  divided. 

2.  If  the  denominator  of  a  fraction  be  midtiplied  or  di- 

(b)  Lat.,  from  denomino,  to  name,  because  it  names  the  parts, 
(c)  Lat.,  from  numero,  to  number. 

*9 


102  FRACTIONS.  [§  114. 

vided,  the  fraction  itself  is  equally  divided  or  multiplied. 
And,  hence, 

3.  If  the  numerator  and  denominator  be  either  both  mul- 
tiplied or  both  divided  by  the  same  number,  the  value  of  the 
fraction  will  not  be  affected. 


REDUCTION. 

§  114.  Let  it  be  required  to  reduce  x  to  a  fraction,  whose 
denominator  shall  be  a. 

We  have,  evidently,  x  =  -  — — -.  §113.3. 

ft  ax 

Otherwise,  1  =-;   .*.  x=. — .  §§42.  c,  113.  1. 

a  a 

Hence,  to  reduce  an  entire  quantity  to  a  fraction 
having  a  given  denominator,  we  have  the  following 

RULE. 

Multiply  the  quantity  by  the  given  denominator,  and 
place  the  product  over  the  denominator. 

3x4      12 
Thus,  to  reduce  3  to  fourths,  we  have  3  =  — - —  =  — . 

4  4 

1.  Reduce  R 2  to  a  fraction  whose  denominator  is  2bc. 

2.  Reduce  52  to  a  fraction  whose  denominator  is 
sin  b  sin  c. 

a.)  If  there  be,  connected  with  the  entire  quantity,  a  frac- 
tional quantity  having  the  given  denominator,  we  may,  ob- 
viously, reduce  the  entire  quantity  as  above,  and  connect 
with  it  by  the  proper  sign  the  numerator  of  the  given  frac- 
tion.   Thus4fc=y-|4=V- 

a2 #2 

1.  Reduce  x to  a  fractional  form. 

x 
a:8_(a3_a:2)      x2_a2_|_;r2       2x^— a2 

Ans. = = • 

x  xx 

x1 a2 

2.  Reduce  x to  a  fractional  form. 

x 


§11 6.]  REDUCTION.  —  LOWER  TERMS.  103 

3.   Reduce   x+y-\ — -    to   a   fractional  form :    also 

x—y 

'—  ±1  and -±1 ;  also x« 

n  q  x 

§  115.   Reduce  —  to  an  entire  form. 
a 

Divide  both  numerator  and  denominator  by  a  (§  113.  3) ; 

ah       , 
then  —  —b. 

a 

Hence,  to  reduce  a  fraction  to  an  entire  or  mixed 

quantity,  we  have  the  following 

RULE. 
Divide  the  numerator  by  the  denominator. 

Note.  If  the  division  is  exact  (§§80.  d,  82.  g),  the  fraction  is 
reduced  to  an  entire  quantity ;  if  not,  the  fraction  can  be  expressed 
in  an  entire  form  by  means  of  negative  exponents  (§  14) ;  or,  if  the 
numerator  be  a  polynomial,  the  fraction  will  be  reduced  to  a  mixed 
quantity. 

a2 — x~ 

1.  Reduce to  an  entire  quantity.    Ans.  a-\-x. 

a — x 

_     _    ,  ,      a4 — x4      1 — cos2v     a2 — x~ 

2.  Reduce  also 


x        lxcosu         a- 

§  116.   To  reduce  a  fraction  to  lower  terms. 

EULE. 

Divide  both  numerator  and  denominator  by  a  com- 
mon divisor.     §  113.  3. 

Note.    To  reduce  to  the  lowest  terms,  we  must,  of  course,  divide 
by  the  greatest  common  divisor  (§  100). 

1.2.3Aa3x5  1.2x 

1.   Reduce  _  .  g  -  A   A  to  its  lowest  terms.  Ans.  -'—. 

3.4.5.6a4a;4  5.6a 

2-  »*"ct<w>»  «»££■»  <»»» 


104  FRACTIONS.  [§117. 

,     „  ,       A*B°—B«~cx"       .    . 

4.  Reduce  -rs — r, Q  ,.  „  to  its  lowest  terms. 

A2cy' — c2x"y" 

Ans.  — :.. 

„     n   ,        nx*-U14-x)"—nxn(l-\-x)n-1 

5.  Reduce ji—. — r-=- to    its    lowest 

terms,  and  simplest  form.  .4ws. 


#3 I 

6.   Reduce  — ~ — —  to  its  lowest  terms. 
X" — 1 


(l+x)*+* 


a~c 


7.    Given  (a-f-a:)  (5-f-a;) — a(b-\-c)  =— ^--J-a:2,  to  find  x. 

^dns.  a:  =  — . 
b 

§  117.  To  reduce  fractions  to  a  common  denominator. 

The  value  of  the  fraction  must  remain  unchanged.  Con- 
sequently, in  effecting  this  reduction,  we  must  either  multi- 
ply the  terms  by  a  common  multiplier,  or  divide  them  by 
a  common  divisor  (§  118.  3).  If  then  the  given  fractions 
be  already  in  their  lowest  terms,  the  common  denominator 
must  be  a  multiple  of  each  of  the  given  denominators. 
Hence,  the  following 

RULE. 

Multiply  all  the  denominators'  together  for  a  new 
denominator,  and  each  numerator  by  all  the  denomi- 
nators except  its  own,  for  a  new  numerator. 

a.)  Otherwise,  Multiply  both  terms  of  each  fraction  by  the 
denominators  of  all  the  other  fractions. 

2  1         5 

1.  Reduce  ^,  -  and  -  to  a  common  denominator. 

o    4  7 

2X4X7_56     1X3X7_21>  5X3X4_60 
3X4X7- 84  ;  4X3X7~"84;  7X3X4~~ 84* 

a  x         oc 

2.  Reduce  T,  -  and  T  to  a  common  denominator. 

be         b 


•  118.]  ADDITION  AND  SUBTRACTION.  105 

aXcXb __ahc      xXbXb  __b2x     xXbXc __   hex 
bXcXb~~b*c  '  'cXbXb~~~b^c'  !>XbXc~~  b2c  ' 

u  1 

3.  Reduce  to  a  common  denominator  —  and  -. 

vJ  v 

x  x 

4.  Reduce  — r-r  and r  to  a  common  denominator. 

a-\-o  a — b 

XXX  X 

5.  Reduce  -,  -,  -,  and  -=  to  a  common  denominator. 

2   o   4  b 

b.)  It  is  evident,  that  the  results  obtained  by  the  rule 
may  often  be  reduced  to  lower  terms,  and  still  have  a  com- 
mon denominator.  This  will  be  obviated  by  taking,  for  a 
common  denominator,  the  least  common  multiple  of  the  de- 
nominators (§  106).  In  that  case,  we  must  divide  the  least 
common  multiple  by  each  of  the  given  denominators,  and 
multiply  the  corresponding  numerator  by  the  quotient. 

XX  X 

0.    Reduce r,  — - — ^r-  and  — r-p  to  fractions  with  the 

a — o    a2 — o2  a-\-b 

least  common  denominator. 


ADDITION  AND  SUBTRACTION. 
RULE. 

§  118.  Reduce  to  a  common  denominator;  and  then 
add,  or  subtract  the  numerators. 

Note.    The  resulting  fractious  in  the  following  examples  should 
be  reduced  to  their  lowest  terms. 

a      c a  ±  c 

b±b~~b~' 

I .    Add  -,  T,  -  and  -. 
a  o   c  r 

a         a+b  a+b 

.  a3-{-bx2  a3-\-bx2 

Am.  — cn—T  —  "V <.  • 

x{a — x){a-f-x)       aJx — x3 

=  what  ?  Ans. 


2(z— 4)       2(x— 2)  a;3— Cx+8" 


106  FRACTIONS.  [§119,120. 

a  3  1 


2  (x—a)  -  ~  4  (x—a)  n  "  4(a;-fa) 
B2x"        —y" 
A2y"~c"—x" 


■=■  what  ? 


6. — — . ,■  =  what  r 


MULTIPLICATION  AND  DIVISION. 

§  119.    To  MULTIPLY  A  FRACTION; 

RULE. 

Multiply  the  numerator,  or  divide  the  denominator, 

by  the  multiplier.     §  113.  1,  2. 

_,        ax  c       ax  ax 

Ihus,  — Xc  =  —  -  = — ;    or,  :=axc~^ic=zaxc~1  =.  — 
c2  ci         c  c 

I    14). 

x       /ax\ 

aX-      (  — ) 

ox              c       \  c  J       ax  ,    ,        ,       ax 

SoTX-  = — r~  = — r~  =7—  5  or,  =«6-1a;e-1=:-r— . 

b    c           b              b         be  oc 

a.)  The  application  of  the  rule  to  the  last  example  gives 

the  common  rule  for  multiplying  fractions  together  ; 

Multiply  the  numerators  together  for  the  numerator  of  the 
product,  and  the  denominators  for  its  denominator. 

Note.    To  multiply  by  —  is  to  multiply  by  "z  and  divide  by  c. 

But   the   fraction   is   multiplied  by  multiplying  the  numerator,  and 
divided  by  dividing  the  denominator. 

,     -«r  i  •  i     aArx  ,      a — x 

1.  Multiply  — —  by  — . 

JL  *C 

n    a+x        x  ,      ,         B*x"     —y"  9 

2.  -Z-x— j— =what?    —  -r^T/X^-7/? 

a2-4-aW-fl9  a— b  .      . 

3. t_ L -X— r-r  =  what? 

:  120.   To  DIVIDE  A  FRACTION  ; 

RULE. 

Divide  the  numerator  or  multiply  the  denominator, 
by  the  divisor.     §  113.  1,  2. 


§  120.]  division.  107 

0 


a  vc/  a 


ab— l       a   .-     " 
brT~e  = =—(§14) 


b   '  b  be'        b   •  c  be 

Also^-c=7^^^(§113-3);  or' 

&X- 

^*  =  aJ-i  jL.a-c-1  —  ab-^x~^c=:~  (§  19.  Cor.  IV.). 
Z»   '  c  ox 

a.)  Hence,  the  common  rule  for  dividing  by  a  fraction  ; 

Invert  the  divisor,  and  multiply. 

Note.    To  divide  any  quantity  by  x  divided  by  c,  is  the  same  as 

to  divide  c  times  that  quantity  by  x. 

b.)  The  last  rule  is  otherwise  demonstrated  thus ; 

a      x a       t\       \  _     fa  m  1\ 

b  +  ~c~b~  \cXX)  -  \b~cJ 


■x. 


-™      a  .   -,       a  a  .   1      ac  rc         .-T_ 

d»  6  0       c        b    *~ 


ac 
bx 


a  .  x fa  .  1\         ac    . 

Apply  the  same  reasoning  to  |  ■—  f- 

c.)  Dividing  either  term  of  a  fraction  has  the  same  effect 
as  multiplying  the  other  term.  Hence,  to  divide  one  frac- 
tion by  another,  we  may  divide  the  terms  of  the  dividend  by 
the  corresponding  terms  of  the  divisor  (i.  e.  numerator  by 
numerator,  and  denominator  by  denominator). 

Note.    This  course  is  convenient,  when  the  divisions  can  be  ex 
actly  performed  (§§  80.  d,  82.  g) ;  and  it  amounts  simply  to  invert- 
ing the  divisor  and  canceling  equal  factors. 

1.    Divide  -z r-  by .         Quot.  —?-. : — „. 

a3 — x3  a — x  a--\-o.x-\-x- 

2  u      a*-p>±x*h    «-*  QuQfl 

a2 — x-  a-\-x 

9       3        ,     ,     10      2,     1.2.3.4.5x5       1.3.5x2  , 

a : —  whit  3 : ? - —  ? 

12  •  4~  IS  •  o"        10.9.8a"      '      9<t»     " 

(-\+    *   W «    -    *   )=what? 

\a — b       a-\-b  /       ^a — b       a-\-b  / 

a2+2ab— b* 


4. 


Ans. 


«2+69 


108  FRACTIONS.  [§121. 

Note.  In  such  examples  ag  the  last,  it  is  generally  most  conven- 
ient to  reduce  the  terms  of  the  dividend  to  a  common  denominator, 
and  also  those  of  the  divisor;  and  then  apply  the  rule  (§120.  a). 

d.)  14--=-;  or  l~xy-i  =  x~iy  (§  18)  —  ^.     Hence, 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 


B2x"  7)  B2 

—  what3     —1—-?-?     1— —  ?    1— P 
A2y"~  '  y'         •  A*'        '  *' 


a 
§  121.  Add  k  to  each  term  of  the  fraction  ■=■•     "Which  is 

a 

.  a       aA-k  „ 

the  greater,  T  or  7  .  7  ? 
b       o-\-/c 

Reducing  to  a  common  denominator,  we  have 

a ab-\-ak  a-\-k ab-{-bk 

b~b*~+Fk  ;  b+k~ b^-bk' 

Now  the  greater  fraction  has  the  greater  numerator. 

But  the  numerators    having  the  term  ab  common,  their 

relative  magnitude  depends  upon  ak  and  bk,  i.  e.  upon  a 

and  b. 

.'.  If  t<1>  then  a  <  b,  ab-\-ak  <  ab-\-bk,  and  y  <   ~!~  . 
o  6      6-j-a; 

If  t->1,  then  a  >  5,  a5-j"a^'  >  «H~^">  an<l  y  >  7T~7~' 

If  -=•  =  1,  then  a  —  b,  ab  -\-ak  =  ab  -}-  &£,  and  7— ,7,  . 
£*  o       b-\-k 

Hence,  k  being  any  positive  quantity, 

a      ^  «+&  ,.  «       ^  , 

r<»  >>  or  =  ,  according  as  T  <,  >,  or  =  1 ;   i.  e.  as 

o  o-f-/c  o 

a  <,  >,  or  —  b. 

That  is,  If  any  positive  quantity  be  added  to  both  the 
terms  of  a  fraction,  the  primitive  fraction  will  be  less,  great- 
er than,  or  equal  to  the  new  fraction,  according  as  it  is  less, 
greater  than,  or  equal  to  unity. 

Add  100  to  the  terms  of  the  fractions  f,  §,  and  f. 


CHAPTER  IV. 


EQUATIONS  OF  THE  FIRST  DEGREE,  CONTAINING 
TWO  OR  MORE  UNKNOWN  QUANTITIES. 


TWO  UNKNOWN  QUANTITIES. 

§  122.  I.)  Let  x-\-y  —  10,  x  and  y  being  both  unknown. 

Here  the  only  condition  (§  38)  is,  that  the  sum  of  the 
unknown  quantities  shall  be  10.  Hence  we  may  have 
;e  =  0,  y=10;  x=zl,  y=9;  x  =.  7,  y  =z 3,  &c. ;  orrcrr 
— 1,  y=ll;  rc:= — 2,  y=:12,  &C  ;  or  y=z — 1,  0,  &C., 
re  =  11,  10,  &c. ;   or  x  —  ^,  y  —  9%;  x  =  — |,  y  =  10f,&c. 

II. .  Again,  let  x — y  =  4. 

Here  the  only  condition  is,  that  the  difference  of  the 
numbers  shall  be  4.  Hence  we  may  have  x  =  4,  y  =  0  ; 
re  =  5,  y  =  1 ;  re  =  7,  y  =  3,  &c. ;  or  a?  =  0,  #  =  — 4,  &c. ; 
or  x  —  20,  y  =  1 6,  &c. 

a.)  Either  of  these  equations  is,  by  itself,  obviously  in- 
determinate (§  109.  c);  and  may  be  satisfied  (§39)  by  any 
one  of  an  infinite  number  of  values  of  x,  with  correspond- 
ing values  of  y. 

b.)  But  the  conditions  may  be  united.  That  is,  it  may 
be  required,  (1.)  that  the  sum  of  two  numbers  shall  be  10, 
and  (2.)  that  their  difference  shall  be  4.  We  shall  then 
have,  at  the  same  time,  rc-f-^  =  10,  and  x — y  =  4.  And 
the  same  values  of  x  and  y  must  satisfy  both  equations. 

c.)  Now,  if  x  —  9,  we  have,  by  the  first  condition,  y  =  1 ; 
and,  by  the  second  condition,  y  =  o.  Thus,  the  same  val- 
ue of  re  satisfies  both  conditions,  but  the  values  of  y  are  dif- 
ferent. Again,  if  y  =.  6,  we  have,  by  the  first  condition, 
x  =  4 ;  and,  by  the  second,  re  —  10.     Here,  the  same  value 

ALG.  10 


110  EQUATIONS.  [§123,124. 

of  y  satisfies  both  conditions,  but  the  values  of  x  are  diffe- 
rent. The  values  in  both  these  cases  are  said  to  be  incom- 
patible. 

d.)  But  the  same  values  of  both  x  and  y  must  satisfy- 
both  conditions  (b).  And,  in  fact,  among  the  values  found 
above  (L,  II.)  there  is  one  set  common  to  the  two  equations, 
viz.  x  =  7,  and  y  =  3  ;  thus  7+3  =  10,  and  7—3  =  4. 

e.)  The  solution  of  the  problem  consists  in  finding  these 
common,  or  compatible  values  of  x  and  y. 

§  123.  The  union  (§  122.  b)  of  the  two  conditions  is  alge- 
braically expressed  by  the  combination  of  the  equations, 
treating  x  and  y  as  symbols  of  the  same  quantities  in  each. 

Note.  It  is  obvious,  that,  if  x  represents  the  same  quantity  in 
the  two  equations,  the  sura  of  x  in  the  first  and  x  in  the  second,  will 
be  2x,  and  their  product,  a:  2.  But  if  x  in  the  second  equation  de- 
noted a  different  quantity  from  x  in  the  first,  it  might  be  distinguished 
■  ',  and  the  sum  of  the  two  quantities  would  be  x+x',  and  the" 
product,  xx1.     The  same  remark,  clearly,  applies  to  y. 

ELIMINATION. 
BY  ADDITION  AND  SUBTRACTION. 

§  124.  Combining  (§  123)  the  equations 

x+y=  10 

x — y  —  4 

adding  them,  member  by  member  (Geom.  §  22),  we  have 

2a;  =  14;    .-.  x  =  7. 

Substituting,  in  the  first  equation  (x+y  —  10),  for  x  its 

value,  we  have 

7+y  =  10;    .-.y=3. 

These  values  of  x  and  y  introduced  into  the  second  equa- 
tion, x— y  =  4,  satisfy  it;  thus 

7—3  =  4,  an  absolute  equation  (§  37.  d). 
,tes.     (1.)    The  value  of  y  might  with  equal  propriety  have 
m  obtained  by  the  substitution  of  x  in  the  second  equation.     Thns 


§  125,  126.]      TWO  UNKNOWN  QUANTITIES.  1  1  I 

7_y  =  4.  ...  y  =  3.  (2.)  If  we  had  subtracted  the  second  equa- 
tion from  the  first,  we  should  have  found  the  value  of  y; and  then,  by 
introducing  it  in  either  of  the  equations,  we  should  find  x. 

§  125.  The  solution  of  the  problem  in  §  124,  it  will  be 
observed,  is  effected  by  removing  one  of  the  unknown  quan- 
tities, till  the  value  of  the  other  has  been  found.  This  is 
called  elimination''  ;  and  the  method  employed  above  is 
called  elimination  by  addition  and  subtraction. 

1.  Given      x-{-y=lo, (1) 

3x+4?/=54 (2) 

.Multiplying  (1)  by  3,  Zx-\-5y  =  45. 

Subtracting  the  la3t  from  (2),         y  —  9. 
Then  from  ( 1 ),        x+9  — 15  ;       .-.  x  =  6. 

2.  Given  Sx— %y  =  —  27,  .         .         (1) 

4z-Hy  =  24.  .         . 

Multiplying  (1)  by  2,  and  (2)  by  3,  we  have 

6as— #  =  —  54,         .         .       (3) 
12x-hy=x72.         .         .  I 

Then,  by  adding  (3)  and  (4), 

18x=18.      .\*=l,y=60. 

Eliminate  x,  by  dividing  (1)  by  3  and  (2)  by  4,  and  subtracting 
iho  first  quotient  from  the  second. 

§  126.  (1.)  When  one  of  the  unknown  quantities  has  the 
same  coefficient  in  both  equations,  it  can  be  eliminated,  if  the 
signs  of  the  equal  coefficients  are  alike,  by  subtraction  ; 
and  if  unlike,  by  addition.     See  §§  57.  18  ;  60.  14. 

(2.)   We  may  cause  one  of  the  unknown  quantities  to 
have  the  same  coefficient  in  both  equations,  by  suitably 
multiplying  or  dividing  one  or  both  of  the  equations. 
1.    Given  5x-\-Gy  =  40, 

Zx-\-2y  =  20,  to  find  x  and  y. 

Ans.  x  =  5,  y  =  2  \. 
(d)  Lat.  elimino,  to  turn  out  of  doors. 


112  EQUATIONS.  [§  127,  128. 

2.  Given  7aH-10y  =  72, 

9x+  3y  =±  63,  to  find  x  and  y. 

3.  Given  2x— dy  =  7, 

x-\-2y  =  14,  to  find  x  and  y. 

BY  COMPARISON. 

§  127.   1.   Resuming  the  equations, 

x-\-y  =  10,        .         .         .         (I) 
*— y  =  4»  ...         (2 

we  have,  from  (1),  x=  10— y, 

and  from  (2),  x  =   4rj-y. 

Equating6  these  values  of  x,  we  have 

10— y  =  4-{-y.    .:  y  =  3,  and  x  ==  7. 
2.    Given  2# — 3y  =  7,  and  a;-J-2y  =  14. 
From  the  first,   %y  =  2x—l ;   .-.  y  ■=.  fa;—  $ . 
From  the  second,   2^=14 — x;     .-.  y=z7 — hx. 
%x — £  =  7 — \x.    .:  x  =  8,  y  =  3. 
In  this  method,  we  find  from  each  equation  the  value  of 
one  of  the  unknown  quantities,  in  terms  of  the  other  un- 
known, and  of  the  known  quantities.      We  equate  these  tws> 
values,  and  from  this  new  equation  find  the  value  of  the  oth- 
er unknown  quantity  ;  and  substitute  as  before. 

Note.  This  is  called  elimination  by  comparison,  because  we 
compare  the  two  values  of  the  unknown  quantity. 

BY  SUBSTITUTION. 

§  128.   1.   Taking  again  the  equations 

x+y=lO,         ...  (1) 

x—y  =  4,  ...  (2) 

we  have,  from  (1),  y  =  10 — x. 

(e)  Lat.  tequo,  to  make  equal.  Quantities  are  said  to  be  equated, 
or  made  equal,  when  they  are  made  to  constitute  the  members  of  an 
equation. 


*  129.]  TWO  UNKNOWN  QUANTITIES.  113 

Substituting  in  (2),  x—  (10— x)  =  4. .-.  x  =  7,  y  =  3. 

2,    Given  2z— 3y  =  7,        .         .         (1) 

x+2y=U.      .         .         (2) 

From  (2),  we  have  y  =  7—lx. 

Substituting  in  (1 ),  2x— 3  (7— \ x)  =  7;  | 

or  2x— 21+|a;  =  7. 

x  =  8,  y  =  o,  as  above  (§127.  2). 

"We  here  ./mrt*,  as  m  the  last  method,  the  value  of  one  of 
the  unknown  quantities  from  one  equation,  and  substitute  it 
in  the  other  equation. 

Notes.  (1.)  This  is  called  elimination  by  substitution.  (2.)  Ei- 
ther of  the  above  methods  may  be  employed  at  pleasure.  Sometimes 
one  will  be  found  most  convenient,  and  sometimes  another.  Practice 
will  enable  one  to  fix  upon  the  best  method  in  each  case.  It  will  be 
useful  for  the  learner,  at  first,  to  solve  each  example  by  all  the  meth- 
ods. 

$129.    1.    Given  %xJr\y  =  11, 

\x-\-\y  =  8,  to  find  x  and  y. 

2.  Given  3z—  4y  =  — 13, 

7x — by  =  0,  to  find  x  and  y. 

Ans.  x=.b,  y—1. 

3 .  Given  —  Sx-{-iy  —  8, 

Sx-\-5y  —  3^,  to  find  x  and  y. 

Ans.  x  —  — £,  y  —  \. 

4.  Given  y  =  2x — 4, 

y  =  — Bx-{-8,  to  find  x  and  y. 

Ans.x=2%,y  —  ?f. 

5.  Given  y  =  ax-\-b, 

y  =  a'x-\-U,  to  find  x  and  y. 

b—V  a'0—aV 

Ans.x=— ,    y  —  — . 

a! — a  a! — a 

With  what  values  of  a,  a',  b  and  V  will  x  and  y,  in  this 
result,  become  zero ?  negative?  infinite  (§  109.  a) ?  inde- 
terminate (§  109.  c)  ? 

*10 


114  EQUATIONS.  [§  129. 

6.  Given  ax-\-by  =  c, 

a'x-\-Vy  =  d,  to  find  x  and  y. 

_  Vc — bd         _ad — a'c 

Ans-x-^ZdV  y-av^Tb- 

7.  Given  x-\-y  =  S,  and  x—y  =  D,to  find  x  and  y. 

Am.  x  =  !(£+■#)>  y  =  K^— D)-     See  §  65.  3. 

8.  A  horse  and  saddle  are  worth  $175 ;  the  horse  is 
worth  six  times  as  much  as  the  saddle.    "What  is  the  value 

of  each  ? 

Solve  the  problem  by  means  of  one,  and  of  two  unknown  quanti- 
ties. 

9.  Let  the  horse  and  saddle  be  worth  a  dollars ;  and  let 
the  horse  be  worth  m  times  as  much  as  the  saddle. 

a  „  ,         , ,.         iua     .        nil 

Ans, value  of  the  saddle  ;  — ; — ,  that  oi  the  horse. 

1+m  l+»» 

10.  A  bill  of  SI 65  was  paid  in  dollars  and  eagles,  the 
whole  number  of  pieces  being  70.  How  many  were  there 
of  each  ? 

Let  x  =  the  number  of  dollars, 

y  =  "  eagles. 

Then         x-\-y  =  70,  and  x+10y  =  1 65. 
Or,  if  x  =  the  number  of  dollars,  then  70 — x  =  the  num- 
ber of  eagles  ;  &c     Or,  let  x  =  the  number  of  eagles ;  &c. 

11.  a  coins  of  one  kind  make  a  dollar,  and  b  of  another 
kind.  How  many  of  each  kind  must  be  taken,  in  order 
that  c  pieces  may  make  a  dollar  ? 

Let  x  =  the  number  of  the  first. 

Then  either  y  or  c — x  will  be  the  number  of  the  second. 

™  x   ,    V       -•  x  ,   c — X        1 

Then  -  +  |  =  l;or-H — 7—  =  1. 

a       b  a         b 

a(e — b)     _  ,      „       ,  .    -     b(a — c)     „  , 
Am.  — — t*-  of  the  first  kind ;  -s — —-  01  the  second. 
a — b  a — 0 

Let  a  =  20,  5  =  10;  and  c  =  12,  13,  20,  10,  21,  9. 
Leta=10,  5=6;  andc  =  8,  9,  10,  6,5,  11. 


§130.]      TWO  OR  MORE  UNKNOWN  QUANTITIES.  115 

a.)   The  nature  of  the  question  requires  whole  number 
for  the  answers.     Such  values,  therefore,  should  be  assign- 
ed to  a,  b,  and  c,  that  the  numerical  values  of  the  above  re- 
sults may  be  integral.     Which  of  the  values  above  comply 
with  this  condition  ? 

b.)  With  what  values  of  a,  b  and  c,  will  the  above  results, 
or  either  of  them,  be  positive  ?  negative  ?  zero  ?  infinite  ? 
indeterminate  ?  How  shall  these  several  results  be  inter- 
preted ? 

12.  Find  a  fraction,  such  that  if  1  be  added  to  its  nume- 
rator, the  value  will  be  |- ;  and  if  1  be  added  to  its  denom- 
inator, the  valiu  will  be  £. 

Let  x  be  the  numerator,  and  y  the  denominator. 

13.  A  certain  number  is  expressed  by  two  digits  whose 
sum  is  9 ;  and  if  it  be  increased  by  five-thirds  of  itself,  tli 
order  of  the  digits  will  be  inverted.    What  is  the  number  ? 

Let  re  =  the  left  hand  digit, 

and  y  —  the  right  hand  digit. 

Then  10x-\-y  =  the  number,  &c. 

14.  A  places  a  sum  of  money  at  interest;   B  invests 
1000  more  than  A,  at  1  per  cent  higher  interest,  and  finds 

his  income  $80  more  than  A's.  C  invests  $1500  more 
than  A,  at  2  per  cent  higher  interest,  and  receives  an  in- 
come greater  than  A's  by  $150.  What  are  the  three  sums 
invested,  and  at  what  rates  ? 

Let  x  =■  A's  sum,  and  y  =■  his  rate  of  interest  per  cent. 

xy 
Then  Too" *"s  mcome?  *»c- 

MORE  THAN  TWO  UNKNOWN  QUANTITIES. 

§130.  I.  Let  x+y+z  =  10  ....  (1), 
x,  y  and  z  being  all  unknown. 

Here  we  may  assign  any  value  we  please  to  any  one  of 
the  unknown  quantities,  and  still  have  an  infinite  number 
of  values  for  the  other  two ;   or  we  may  assign  any  values 


116  EQUATIONS.  [§18L 

whatever  to  two  of  them,  and  find  a  corresponding  value  for 
ihe  third.     Thus,  the  problem  is  doubly  indeterminate. 

II.  Again,  let  2x— y-\-3z  =  7    ...        (2) 
This  equation  is  equally  indeterminate  as  the  first.    And 

if  we  unite  the  two  conditions,  W6  may  still  assign  any  val- 
ue we  please  to  one  of  the  unknown  quantities,  and  deduce 
corresponding  common  values  for  the  other  two.  Or,  elim- 
inating one  of  the  unknown  quantities,  we  shall  have  a  sin- 
gle equation  with  two  unknown  quantities. 

Thus,  adding  (1)  and  (2), 

8x+±z  =  17         .         .         .     (a) 

Hence,  the  problem  is  still  indeterminate  (§  122.  a). 

III.  But  again,  let 

3x+2y+4z=z27        .        .        .     (3) 

Multiplying  (1)  by  2, 

2x+2y+2z  =  20         .         .         .(b) 

Subtracting  (b)  from  (3), 

x+2z  =  7  .         .         .(c) 

Combining  (a)  and  (c),  [§  123],  we  find  x=z3,  y  =  5,  and, 
then  substituting  in  (1),  (2)  or  (3),  z  =  2. 

Note.    We  might,  obviously,  have  employed  either  of  the  other 
methods  of  elimination  (§  124-128). 

§  131.  Hence,  to  find  the  common  values  of  three  un- 
known quantities,  from  three  equations,  we  eliminate  one  of 
the  unknown  quantities  from  all  the  equations,  thus  forming 
two  equations  with  two  unknown  quantities.  We  then  solve 
these  equations  by  §  124-128. 

1.  Given  &H4rK*  =  12, 

*—  y+  z  =  12> 

2x-\-3y — 4s  =  12,  to  find  x,  y  and  z. 

2.  Given  x-\-  ly+^s  =27 

x-\-\y-\-\z=z  16,  to  find  x,  y,  and  z. 

Ans.  x  =  1,  y  =  12,  z  =  60. 


§  182-134.]     SEVERAL  UNKNOWN  QUANTITIES.  117 

§  132.  "We  have  found  one  equation  with  tivo  unknown 
quantities,  and  one  or  two  equations  with  three  unknown 
quantities  to  be  indeterminate.  In  like  manner,  if  we  had 
three  equations  with  four  unknown  quantities,  by  eliminat- 
ing one  of  the  unknown  quantities,  we  should  have  two 
equations  containing  three  unknown  quantities,  and,  of 
course,  indeterminate.  By  like  reasoning,  we  shall  find, 
that  any  number  whatever  of  equations  must  be  indetermin- 
ate, if  the  number  of  unknown  quantities  is  greater  than  the 
number  of  independent  equations. 

Notes.  (1).  Independent  equations  are  those,  of  which  no  one 
is  implied  by  the  rest.  Thus  x+y=B,  and  3x+3y  =  9  are  not 
independent,  because  one  is  a  necessary  consequence  of  the  other. 
(2.)  When  a  number  of  equations  containing  several  unknown 
quantities  are  spoken  of,  they  must  be  understood  to  be  independent 
equations,  unless  the  contrary  is  stated,  or  clearly  implied  by  the 
connection. 

§  133.  If  we  have  four  equations  involving  four  un- 
known quantities,  the  elimination  of  one  of  the  unknown 
quantities  will  result  in  three  equations  containing  three  un- 
known quantities,  which  may  be  solved  by  §  131.  The 
same  reasoning  will  obviously  extend  to  any  number  of 
equations  containing  an  equal  number  of  unknown  quanti- 
ties. 

§  134.  Hence,  to  find  the  value  of  any  number  ol 
unknown  quantities  from  an  equal  number  of  equa- 
tions, we  eliminate  one  of  the  unknown  quantities  from 
all  the  equations,  thus  diminishing  by  one,  at  the  same 
time,  the  number  of  equations  and  of  unknown  quan- 
tities contained  in  them. 

We  then  eliminate,  from  the  new  equations,  another 
unknown  quantity,  and  so  on,  till  we  arrive  at  a  sin- 
gle equation  containing  one  unknown  quantity. 

1.    Given  7x—2z+3u  =  17, 
4&—2z+t  =  ll, 
5y—3x—2u=:8, 


•118 


EQUATIONS.  [§  134. 


Ay— 3u+2t=0, 

3z-\-8u  —  33,  to  find  x,  y,  z,  u  and  t. 
Ans.  x  =  2,y  =  4:,z  =  3,u=z3,t=zl. 

2.  Given  2x—3y+2z  =  13, 

2u — x=.  15, 
2y+z  =  7, 
5y-\-3u  —  32,  to  find  x,  y,"u  and  z. 

Ans.  x  =  3,  y=l,  m  =  9,  z  =  5. 

3.  The  sum  of  four  numbers  is  25.  Half  of  the  first 
number  is  equal  to  twice  the  second,  and  to  three  times  the 
third;  and  the  fourth  is  four  times  the  third.  What  are 
the  numbers  ? 

Let  u,  x,  y  and  z  represent  the  numbers. 
Also  let  x  represent  one  of  the  numbers,  and  solve  the 
problem  with  one  unknown  quantity. 

4.  Find  three  numbers,  such,  that  the  sum  of  the  first 

and  second  shall  be  15  ;  the  sum  of  the  first  and  third,  16  : 

and  the  sum  of  the  second  and  third,  17. 

Solve  the  above  problem  by  one,  by  two,  and  by  three  unknown 
•quantities. 

5.  A,  B  and  C  form  a  partnership.  A  contributes  a  cer- 
tain sum ;  B  contributes  a  times,  and  C,  b  times  as  much 
as  A ;  and  the  whole  stock  is  c.  How  much  did  each  con- 
tribute?    See  §55.  4. 

c  etc 

■Ans.  — — —,  A's  part;    ,  ,      ,  ,,  B's  part,  &c. 
I-\-a-f-b  l-\-a-j-o 

5.  A  and  B  can  perform  a  piece  of  work  in  8  days ;  A 
and  C,  in  9  days ;  and  B  and  C,  in  10  days ;  in  how  many 
lays  could  each  person,  alone,  perform  the  same  work  ? 

Let  x,  be  the  number  of  days  required  by  A ;  y,  by  B ; 
and  z,  by  C. 

Then,  in  1  day,  A  will  perform  -  of  the  work  ;    B,  -  ; 

*  y 

md  C.  -.     But  A  and  B  together  perform,  in  1  day,  -  of 

rhe^work;  &c 


§135,  136.]     SEVERAL  UNKNOWN  QUANTITIES.  119 

1,1        1        1,1        1  ,  1,,1_1 

x      y      8     $x      z       9  y      z      10 

Note.    Instead  of  clearing  of  fractions,  regard  — ,  —  and  —  as  the 

x    y  z 

unknown  quantities;    and  from  tbeir  values,  when   found,  find  the 
values  of  x,  y  and  z  (§  50). 

Ans.  A  in  14§§  days;  B,  in  17|| ;  and  C,  in  1337T. 

6.   Let  A  and  B  perform  the  work  in  a  days ;   A,  and 

C,  in  b  days  ;  and  B  and  C,  in  c  days  ;  and  find  the  general 

expression  for  the  time  in  which  each  person,  alone,  would 

perform  the  work. 

.  2abc  . ,     .  2abc         _, 

Ans.  -= — ; 7,  A  s  time  ;  -= — ; — 5 ,  B  s ;  and 

oc-\-ac — ao  oc-\-ao — etc 

2abc         „. 

~rr a-'  Cs- 

ab-\-ac — be 

§  135.  We  have  seen,  that,  when  the  number  of  unknown 
quantities  is  greater  than  the  number  of  independent  equa- 
tions, the  problem  is  indeterminate.  "When,  on  the  other 
hand,  the  number  of  unknown  quantities  is  less  than  the 
number  of  independent  equations,  the  equations  are  incon- 
sistent in  their  conditions,  and  cannot  all  be  satisfied  by  the 
same  values  of  the  unknown  quantities.  For,  if  the  values 
found  from  two  equations  containing  two  unknown  quanti- 
ties would  satisfy  a  third,  this  would  be  implied  by  the  rest, 
and,  of  course,  would  not  be  independent  of  them  (§  132. 
N.  1).  E.  g.  the  equations  x-\-y=.  10,  x — y  —  4,  and  2x-{-y 
=  40,  are  obviously  inconsistent. 

§  136.  When  a  single  equation  containing  more  than  one 
unknown  quantity  is  considered  by  itself,  the  unknown 
quantities  are  frequently  called  variables  ;  and  one  of  them 
is  said  to  be  a  function  (§  26)  of  the  rest. 

a.)  Thus,  in  the  equations  2y-\-3x  =  10,  y  =  ax-\-b,  y  is 
a  function  of  'x,  and  x  of  y ;  or,  as  it  is  usually  expressed, 
y  —  F(x),  and  x  =  F(y).  For,  if  we  give  any  value  what- 
ever to  one  of  these  quantities,  we  can  deduce  a  correspond- 
ing value  for  the  other ;  and,  if  we  vary  the  value  of  the  first, 
the  value  of  the  second  undergoes  a  corresponding  change. 


120  EQUATIONS.  [§  137. 

h.)  If  an  equation  contain  more  than  two  unknown  quan- 
tities, each  of  them  is  a  function  of  all  the  rest.  Thus,  in 
the  equation  2x-\-3y-\-z  =  75,  we  have  x  ==  F(y,  z)  [i.  e.  x 
a  function  of  y  and  z],  y  —  F(xy  z),  and  z  =  F(x,  y). 

Notes.  (1.)  Of  two  quantities,  that  of  which  the  other  is  said 
to  be  a  function,  is  called  the  independent  variable.  (2.)  Either  of 
the  unknown  quantities  may,  obviously,  be  made  the  independent 
variable,  and  the  other  will  be  the  function.  (3.)  If  there  are  more 
than  two  variables,  one  may  be  regarded  as  a  function  of  all  the  rest, 
they  being  all  independent;  or  one  may  be  a  function  of  the  second, 
the  second,  of  the  third,  and  so  on,  the  last  only  being  independent. 

§  137.  Arithmetic,  in  its  ordinary  applications,  furnishes 
only  positive  and  definite  solutions.  It  is,  therefore,  some- 
times said,  that  negative,  infinite  and  indeterminate  results 
do  not  furnish  a  proper  answer  to  a  question.  The  answers 
which  they  furnish  would  indeed  not  be  intelligible  to  one 
unacquainted  with  the  algebraic  language.  But  to  one  fa- 
miliar with  that  language  a  negative  result  answers  a  ques- 
tion as  directly  and  intelligibly  a3  a  positive  ;  an  infinite,  as 
a  finite.     See  §§  4.  9,  10 ;  109.  10,  16. 

Thus,  when  we  inquire,  how  long  it  will  be  before  a  cer- 
tain event  will  take  place,  we  equally  answer  the  question 
by  saying  that  it  will  take  place  in  12  years  (§  109.  14),  or 
in  no  years  (i.  e.  now,  §  109.  15),  or  that  it  took  place  4 
years  ago  (§  109. 16),  or  that  it  will  never  take  place  (§  109. 
9,  17),  or  that  it  is  taking  place  all  the  time  (§  109.  c). 

In  like  manner,  if  we  inquire  how  far  east  a  certain  point 
lies,  we  equally  answer  the  question  by  saying  an  infinite 
distance,  a  finite  distance  (as  10  miles),  no  distance,  a  dis- 
tance west,  or  that  such  a  point  exists  every  where  in  a  line 
running  east  and  west. 

Arithmetic  does  not  ordinarily  take  cognizance  of  infin- 
ite or  indeterminate  results ;  and,  regarding  numbers  sim- 
ply as  such,  without  respect  to  their  character  as  positive 
or  negative  (§  8),  its  questions  must  be  proposed  in  such  a 
manner,  that  an  answer  may  be  expressed  by  a  number 
simply. 


§  138-140.]     SEVERAL  UNKNOWN  QUANTITIES.  121 

§  188.  It  will  be  observed,  that  between  the  positive  and 
negative  values,  we  always  have  a  value  equal  to  zero  or 
to  infinity,  i  e.  equal  to  0  or  %.  See  §§  4.  6-10 ;  109.  8-10, 
b,  14-16.  That  is,  between  the  positive  and  negative  re- 
sults, there  is  one,  either  equal  to  0,  or  whose  denominator 
has  become  0. 

§  139.  We  have  %  —  cc  (§  109.  a).     That  is, 

(1.)  A  finite  quantity  divided  by  zero  is  equal  to  infinity. 

Also  (§  42.  c),  a  =  OX  co.     That  is, 

(2.)  Zero  multiplied-  by  infinity  is  equal  to  a  finite  quan- 
tity. 

Again  (§  42.  d),  —  =  0.     That  is, 

CO 

(8.)  A  finite,  divided  by  an  infinite  quantity  is  equal  to 
zero. 

Note.  We  arrive  at  the  idea  of  infinity  by  continually  diminish- 
ing a  divisor,  and  thus  finding  a  greater  and  greater  quotient  (§  109. 
a).  Hence  0  is  sometimes  said  to  denote  an  infinitely  small  quan- 
tity, or  an  infinitesimal  (i.  e.  a  quantity  less  than  any  assignable 
quantity  [§  109.  a]). 

§  140.  The  expression  $  is  not  always  indeterminate. 
For,  instead  of  the  whole  numerator  and  denominator,  a 
common  factor  may  have  been  reduced  to  0.  If  then  this 
common  factor  be  removed  (§  113.  3),  the  expression  will 
no  longer  be  indeterminate.  Thus,  when  b  =  a, 
a2— b*      0      _      «2_S3      (a+J)(a— J) 

r=7v     But -—  v  ^  JK  — J-=a-\-b  —  2a. 

a — b       0  a — b  a—b  ' 

when  b=.a. 

oji jf      o 

So,  if  bz=z  a, r—7;-     But,  performing  the  division. 

a — b       0 

ft« jf 

and  then  making  b=i  a,  we  have -z^na"-1  (§  96.  b). 

a — b 

xs i 

!•   - — —  =  what,  when  x  =  1  ?  Ans.  3. 

x— 1 

2.    -z z-  =  what,  if  x  =  y  ?  Ans.  —. —  =  0. 

x~—y2  v  x+y 

ALG.  11 


122  INEQUALITIES.  [§141-144 

o.   -. ^0  =  what,  if  x  =  y  ?  ^tns.  — ^L  =  o>. 


CHAPTER  V. 


INEQUALITIES. 


§  141.  Two  quantities,  connected  by  the  sign  <  or  >  (§  2. 
b),  constitute  an  inequality.  An  inequality  may  be  cal- 
1  ed  increasing,  or  decreasing,  according  as  the  second  mem- 
ber is  greater  or  less  than  the  first.  When  two  inequalities 
both  increase,  or  both  decrease,  they  may  be  said  to  have  the 

ne  tendency,  or  to  subsist  in  the  same  sense  or  direction  ; 
otherwise,  they  are  of  contrary  tendency. 

§  142.  Operations  upon  inequalities  are  similar  to  those 
upon  equations,  and  depend  chiefly  upon  an  analogous  ax- 
iom (§  42) ;  viz. 

Unequal  quantities,  equally  affected,  remain  unequal. 

Hence,  if  equal  quantities  be  (1.)  added  to,  or  (2.)  sub- 
tracted from,  both  sides  of  an  inequality,  or  if  both  sides  be 
(3.)  multiplied,  or  (4.)  divided  by  equal  quantities,  the  results 
vdll  be  unequal. 

§  143.  In  transforming  an  inequality,  however,  we  must 
not  only  preserve  the  inequality,  but  we  must,  at  every  step, 
determine  which  way  it  tends  (i.  e.  which  member  is  the 
greater).  Hence,  the  necessity  of  observing  the  following 
obvious  principles. 

§  144.   a.)   If  equal  quantities  be  added  to,  or   sub- 
tracted from,  both  members  of  an  inequality,  the  tenden- 
cy of  the  inequality  will  always  remain  unchanged.     Thus, 
10>6;  10±8>6±8 


145-147.]  INEQUALITIES.  123 

— 10<— 6;  — 10±12<— 6±12.     See  §  6.  a. 

Note.  Hence,  transposition  applies  to  inequalities,  in  like  man- 
ner ae  to  equations.     Thus, 

10— 5>12— 8.   .-.  10>17— 8. 

So,  ify3+*2— E2>0,  theny2-fa:2>i22,  andy3>i22— x*. 

§  145.  b.)  "With  still  greater  reason, 

If  two  inequalities,  having  the  same  tendency,  be  added, 
nember  by  member,  there  xvill  result  an  inequality  of  th 
same  tendency. 

Thus,  9>7  and  — 1>— 3.  .-.  8>4. 

So  if  a~>b,  and  m>n,  then  a-\-m>bA^n. 

Note.  If  one  inequality  be  subtracted,  member  by  member,  fr«fn 
■nother  of  the  same  tendency,  the  result  will  not  always  be  an  ine- 
quality ;  nor,  if  it  be,  will  it  necessarily  have  the  same  tendency. 

§  146.  c.)  If  the  members  of  an  inequality  be  subtracted 
from  the  same  member,  the  tendency  of  the  inequality  will  be 
"hanged.     Thus, 

8>6,  and  10— 8<10— 6. 

In  like  manner,  0 — 8<0 — 6  (i.  e.  — 8< — 6).     Hence, 

d.)  If  the  signs  of  both  members  be  changed,  the  tendency 
will  be  changed. 

Note.  This  results  directly  from  the  principle,  that,  of  negative 
quantities,  that  which  is  numerically  the  greatest  is  absolutely  the 
least  (i.  e.  leaves  the  least  remainder).     See  §  6.  a. 

§  147.  e.)  If  both  members  of  an  inequality  be  multi- 
plied or  divided  by  the  same  positive  number,  the  re- 
sulting inequality  will  have  the  same  tendency ;  if  by  the 
same  negative  number,  the  tendency  xoill  be  changed. 
Thus, 

6>— 8 ;  and  6X3>— 8x3,  or  18>— 24. 
But  6X—  3<— 8X—  3,  or  — 18<-f  24. 

So                    6-^2>— 8-r-2,  or  3>— 4. 
But  6^ 2<— 8-i 2,  or  — 3<+4. 

Also,  if    a >  b,  ah  >  bk  (§  121),  but  —  ah  <—  bt 


!24  POWERS  AND  ROOTS.  [§148-151- 

§  148.  /.)  Hence,  an  inequality  may  always  be  cleared 
of  fractions.  For,  if  we  multiply  by  a  positive  denomina- 
tor, the  tendency  remains  the  same ;  if  by  a  negative,  it  is 
changed.  Or,  if  the  denominator  is  negative,  we  may  place 
its  sign  before  the  fraction,  and  then  multiply  by  the  posi- 
tive denominator  (§§  68.  b,  80.  b). 

§  149.  g.)  If  the  members  of  an  inequality  be  positive, 
and  be  both  raised  to  the  same  positive  integral  power 
of  any  degree  whatever,  the  tendency  of  the  inequality  tvill 
remain  unchanged. 

Thus,  7>3;  72>32; 

Note.  This  holds  equally  of  fractional  powers  or  roots  (§23. 
/>),  so  long  as  we  confine  ourselves  to  their  positive  values  (23./.  1). 
If  we  regard  the  negative  values  of  an  even  root,  the  tendency  is,  of 
course  changed. 

§  150.  h.)    Whatever  be  their  signs,  if  the  members  of  an 
inequality  be  both  raised  to  the  same  odd  positive  potver. 
the  tendency  will  remain  unchanged.     Thus, 
— 3<2;  (— 3)3<23« 


CHAPTER  VI. 


POWERS  AND  ROOTS. 


MONOMIALS. 


§  151.  To  raise  a  monomial  to  any  power; 
Multiply  the  exponent  of  each  factor  by  the  expo- 
nent  of  the  required  power.  (§24.  d). 

a.)  This  rule  depeuds  on  the  obvious  principle,  that  a 


§  151.]  MONOMIALS.  125 

power  of  a  product  is  equal  to  the  product  of  the  same  pow- 
ers of  the  several  factors.     Thus, 

{abc)n—anbncn\  [pbc)*  =  a2b2c2  ;  («J)^a¥. 
b.)  This  rule  applies  equally  to  numerical  and  literal 
factors ;  and,  so  far  as  Algebra  is  concerned,  it  is  sufficient. 
It  is  proper,  however,  to  perform  upon  the  numerical  coef- 
ficient the  arithmetical  operations  indicated  by  its  expon- 
ent. Thus,  if  its  exponent  be  positive  and  integral,  raise 
the  coefficient  to  the  arithmetical  power  denoted  by  the  ex- 
ponent ;  if  the  exponent  be  positive  and  fractional,  raise  the 
coefficient  to  the  power  denoted  by  the  numerator  and  ex- 
tract the  root  denoted  by  the  denominator ;  if  the  exponent 
be  negative,  perform  the  same  operations  as  if  it  were  pos- 
itive, and  place  the  result  in  the  denominator  of  a  fraction, 
of  which  the  other  factors  of  the  monomial  constitute  the 
numerator. 

c.)  The  sign  of  an  even  integral  (§  22.  d)  power  is  posi- 
tive ;  the  sign  of  an  odd  integral  power  is  the  same  as  thai 
of  its  base  (§  22.  N.).     See  §  11.  N.  2. 

1.  What  is  the  fourth  power  of  2ab^x^y~^? 

Ans.  2*a*b*x%~%=  16a*h*x*y~$. 

2.  {—Za-b- 1)3  —  what?      (— aWcx)2?      (aV3)2? 

3.  (na-2x~?)^  =  whsLt?  (a~*x~%)~*?  (10&)6? 
(anxr)t?     (^)s-i? 

-  -  -  i  r 

4.  (anx,J)s  =  Yfhsit?    (3a-2b*x2y~?)   5?    (fi2x-2)~%  ? 

d.)  In  determining  the  sign  of  a  fractional  (§  22.  d)  pow- 
er, its  exponent  should  be  reduced  to  its  lowest  terms. 
Then,  if  the  numerator  of  the  exponent  is  an  even  number, 
the  power  is  positive;  if  the  denominator  is  even,  the  pow- 
er of  a  positive  quantity  is  ambiguous  (i.  e.  ±),  and  of  a 
negative  quantity,  imaginary  (§  23./) ;  and  if  both  numera- 
tor and  denominator  are  odd  numbers,  the  power  has  the 
same  sign  as  the  quantity  itself. 

*ll 


126  POWERS  AND  ROOTS.  [§  152. 

e.)  A  power  of  &  fraction  is  found  by  raising  both  nume- 
rator and  denominator  to  the  required  power  (§§  119.  a,  120. 
c).  Or,  all  the  factors  of  the  denominator  may  be  carried 
into  the  numerator,  if  we  change  at  the  same  time  the  signs 
of  their  exponents  (§§14,  17)  ;  and  then  the  quantity  may 
be  treated  like  any  other  monomial. 


G)': 


ROOTS. 


§  152.  From  the  preceding  rule  (§  151),  we  deduce  the 
following  specific  rule,  in  which  the  term  root  is  used  in 
the  same  6ense  as  in  Arithmetic. 

To  extract  any  root  of  a  monomial  ; 

Extract  the  root  of  the  numerical  coefficient  as  in  Arith- 
metic ;  and  divide  the  exponent  of  each  literal  factor  by  the 
number  of  the  root. 

a.)  This  rule  is  obviously  included  in  the  preceding 
(§  25).  But  for  convenience,  and  on  account  of  the  rery 
frequent  necessity  of  extracting  the  square  and  cube  roots, 
it  is  given  here  in  a  distinct  form. 

b.)  An  odd  root  of  a  positive  quantity  is  positive;  of  a 
negative  quantity,  negative  (§  23.  e). 

c.)  An  even  root  of  a  2}ositive  quantity  is  either  positive 
or  negative  (§  23./.  1). 

d.)  An  even  root  of  a  negative  quantity  is  imaginary 
(22./.  2). 

3 

1.  "What  is  the  square  root  of  2oaHc-yxz'? 

Ans.  J{25a*bc-ix3)=  (25a2bc~^x^  —  5a5-c~M. 

2.  y(49a3J2a:-3)=what?    (100«-45ma;2M)- ?    (x^? 

3.  V-^  =  what?    (*£)*?    fe£L)*P 

6464y12  \2(bzzJ  \**/c*/r/ 


[§  153-150.  KAD1CALS.  127 

§  153.   Any  root  of  any  monomial  can  be  algebraically 

expressed,  but  it  is  not  always  possible  to  perform  exactly 

the  arithmetical  operations  upon  the  coefficient.     Thus  the 

jl  1 
square  root  of  2ab-  is  2'2a2b;  but  the   exact   arithmetical 

computation  of  2  2  cannot  be  attained.  Such  a  root  is  cal- 
led incommensurable^  irrational0  or  surdh.  A  numerical 
quantity  whose  root  can  be  exactly  found  is  called  a  •per- 
fect power. 

Note.  It  will  be  shown  hereafter,  that,  if  a  root  of  a  whole  num- 
ber is  not  a  whole  number,  it  cannot  be  expressed  at  all  except  by 
approximation. 

§  154.  The  use  of  the  term  perfect  power,  as  applied  to 
algebraic  monomials,  is  sometimes  restricted  to  the  cases  in 
which  the  numerical  coefficient  is  a  perfect  power,  and  each 
exponent  is  divisible  (<§  80.  d)  by  the  number  of  the  root. 
The  roots  of  all  quantities  which  are  not  perfect  powers 
are  called  irrational,  radical  (§  23.  d.  N.)  or  surd  quantities. 

§  155.  A  radical  quantity  can  frequently  be  reduced  to  a 
s  impler  form.     Thus, 

(192a3fr3c)S=(64a352X3ac)^  =  8a5(3ac)^. 

(I08a^x)^=  (27a3£6X4aVp  =  3a52(4a2x)*. 
We  here  separate  the  root  into  two  factors,  one  of  which 
is  rational  (i.  e.  expressed  by  integral  exponents),  while  the 
other  is  radical  (i.  e.  expressed  by  fractional  exponents). 
This  can,  obviously,  be  done,  whenever,  after  the  extrac- 
tion of  the  root,  any  of  the  exponents  are  improper  frac- 
tions; or,  when,  before  the  extraction,  any  of  them  are 
greater  than  the  number  of  the  root,  and  not  exact  multi- 
ples of  it. 

§  156.   We  shall,  evidently,  effect  this  simplification,  it, 

(/)  Lat.  in,  not,  con,  together  and  mensura,  measure;  having 
520  common  measure  (§100)  with  unity,  (g)  Lat.  in,  not  and  ra- 
tio, relation,  ratio ;  whose  ratio  to  unity  cannot  be  exactly  expressed, 
(/i)  Lat.  surdus,  that  is  not  heard;  because  it  cannot  be  expressed. 


128  P0WER9  AND  ROOTS.  [§  157,  158. 

in  extracting  the  root,  we  divide  the  exponent  of  each  let- 
ter by  the  number  of  the  root,  and  set  the  integral  part  of 
the  quotient  as  an  exponent  of  the  letter  in  one  factor,  and 
the  fractional  part  as  an  exponent  of  the  same  letter  in 
another  factor.  If  the  root  has  been  extracted,  we  have 
only  to  reduce  all  the  improper  fractions  among  the  expo- 
nents to  mixed  numbers,  and  set  each  letter  under  its  integral 
exponent  in  one  factor,  and  under  its  fractional  exponent 
in  another. 


1.  Eeduce  y60a364a;  to  its  simplest  form. 

Ans.  (60aH*x)^=:  (4.l5)%aah*x^=2ab*(ttaxft= 

2ab"  +/loax. 

2.  Reduce  (7oa2b5x7)2  to  its  simplest  form. 


3.  Reduce  also  3^/5 4a8 x3  ;    ^/32x2y5 ;    *Jasbpc~2x; 
J(2p)x2  ? 

4.  Separate  a2b3c4x    »    into  rational  and  radical  factors. 

CT  fin     _^n       3r>      JL2>'» 

Ans.  aWcxXahh*xn=aWcxXa^nb12V-nxi'-n  = 

a2b2cx  l  aV«6"&4"<^12"'—  a*b*cz{aanb**c*nxi  ■»)A» 
See  §  1G0. 

§  157.  a.)  In  simplifying  an  irrational  fraction,  it  is  gen- 
erally best  to  multiply  both  numerator  and  denominator  by 

a  multiplier   which  will  make  the  denominator  rational. 

/3\- 

Thus,  we  may  simplify  the  fraction  f-J  2,  as  follows : 

/8\i_3*_**7*_(8J7)*_l,sla 

W    -^-^|~~7~~7(^  * 

Note.    If  the  sum  of  the  exponents  of  each  letter  in  two  mono- 
mials be  an  integer,  the  product  will,  of  course,  be  rational. 

§  158.  b.)  Every  negative  quantity  can,  obviously,  be  re- 


§159-161.]  RADICALS.  129 

garded  as  containing  the  factor,  — 1,  together  with  a  posi- 
tive factor. 

Thus—  a  —  a{— 1)  ;  —  a2  =  a2(— 1)  ;  — 25  =  25(— 1). 

Hence,  (— a)  2  —  a2(— 1)^ ;  (— a2)  2  =  (a2)  2(— 1)^  = 
a*/ — 1. 

1.  (— ^2)2"=what?  Am.  Bj— 1. 

2.  (— 2o«2^)2"  =  what?  Ans.halPx^J—\. 
Hence,  every  even  root  of  a  negative  quantity  consists  of 

a  real  quantity  multiplied  by  */ — 1. 

Note.  Such  expressions  as  the  above  must  not  be  regarded  as 
having  any  actual  value  whatever.  One  factor  is  real,  but  the  other 
is  imaginary;  and  the  product  is,  of  course,  imaginary. 

§  159.  Addition,  subtraction,  multiplication  and  division 
are,  of  course,  performed  upon  irrational  quantities  accord- 
ing to  the  general  rules.  In  addition  and  subtraction,  it  is 
frequently  more  convenient  to  separate  the  quantities  into 
their  rational  and  radical  factors,  and  reduce  the  resulting 
polynomials  by  §  33.  c. 

§  1 60.  After  the  separation  of  the  rational  and  radical 
factors  of  a  monomial,  it  is  frequently  convenient  to  reduce 
all  the  fractional  exponents  to  a  common  denominator,  and, 
writing  only  the  numerator  of  each  exponent  over  its  letter, 
enclose  the  whole  in  a  parenthesis  under  the  reciprocal  of 
the  denominator  as  an  exponent;  or,  if  preferred,  place  the 
whole  under  a  radical  sign  with  the  common  denominator 
over  it.     See  §  156.  4. 

Notes.  (1.)  Radicals  which  have  the  same  quantities,  both  nu- 
merical and  literal,  under  the  same  fractional  exponent  or  radical 
sign,  are  called  similar  radicals.  (2.)  The  rational  factor,  multi- 
plied by  a  radical,  is,  of  course,  properly  called  the  coefficient  of  the 
radical. 

§  161.  The  rational  factors  may  be  placed  under  the  rad- 
ical exponent  or  sign,  if  their  exponents  be  reduced  to  frac- 
tions having  the  common  denominator.     This  is  commonly 


1#0  POWERS  AND  ROOTS.  [§  162. 

called  carrying  the  coefficient  of  the  radical  under  the  sign. 
Thus, 

2     1  i 

xja=z&«p  =  (ax-y,  or  J  (ax-). 

1.  x(2JRxy  =  what?      Ans.  (2Rx*y,  or  J (2Rx*). 

2.  x(2ifo)^:=:what?    Ans.  (2Rx*)$,  or  V(2ifa4). 
This  transformation  is  particularly  useful  in  finding  an 

approximate  root  of  a  number.     Thus, 

7y5  =  7X2  (the  nearest  unit)  =  14.     But 

7^5  =  7*5*=  (7*.5)*=  (49.5)*=  (245)^  =  16  (the 
nearest  unit). 

Note.  In  extracting  the  root  of  5  and  multiplying  by  7,  we  mul- 
tiply the  error  in  the  root  by  7.  In  the  other  process,  we  avoid  this 
source  of  inaccuracy. 

Remark.  In  some,  especially  of  the  earlier  treatises,  the  radical 
sign  is  used  almost  to  the  exclusion  of  fractional  exponents.  The 
exponent,  however,  is  much  more  convenient,  and  many  of  the  diffi- 
culties connected  with  the  calculus  of  radicals,  as  it  is  called,  dis- 
appear, when  the  exponent  takes  the  place  of  the  sign.  Hence,  if  it 
is  intended  to  u«e  the  radical  sign  in  expressing  the  result,  it  is  still 
generally  best  to  employ  the  exponent  in  the  operations  by  which 
the  result  is  obtained. 

IMAGINARY  QUANTITIES. 

§  162.  The  expression  */ — 1  may  be  taken  as  the  repre- 
sentative of  all  imaginary  quantities.  The  treatment  of 
imaginary  quantities  will  be  best  illustrated  by  considering 
soine  of  the  powers  of  J — 1.     Thus, 

(y-i)2  =  (-i)^.(-i)^=-i. 

(y-l)3  =  (-l)^-_l)*  =  ^l(-l)*  =  -^-l. 

k/-l)*=(-l)*=(-l)»  =  l. 
U-l)o-^_l.    (y_l)G-_i;    (y_l)7__v/_1; 

(^-1)8  =  1;    &c. 

Hence,     (,/— a2)2  =  (aj—l)°  =  a8X— 1  =  —a2. 
y—a°  y— 53  =  ay— lx  V— 1  =  abX— 1  =  — ab. 


§163,  164.]      POLYNOMIALS. BINOMIALS.  131 

Notes.  (1.)  Caro  must  be  taken  not  to  confound  imaginary 
with  irrational  expressions.  A  numerical  surd,  as^/2,  cannot  be 
exactly  expressed  in  units  or  parts  of  a  unit,  but  we  may  approximate 
as  near  as  wo  please  to  its  true  value.  An  imaginary  expression, 
on  the  other  hand,  as  ^/ — 1,  has  no  actual  value,  and  we  can,  of 
course,  make  no  approach  to  its  value;  nor  can  one  quantity  be  said 
to  come  any  nearer  to  its  true  value  than  another.  Thus,  no  quan- 
tity can  be  conceived,  which,  multiplied  into  itself,  will  produce — 1; 
and  the  expression  / — 1  is  merely  a  symbol  of  an  impossible  opera- 
tion; a  symbol,  to  which  there  exists  no  corresponding  quantity. 
(2.)  It  may  be  thought,  that  such  symbols,  not  representing  quanti- 
ty, can  be  of  no  utility,  and  should  have  no  place  in  investigation? 
relating  to  quantity.  But  some  of  the  most  remarkable  and  useful 
results  of  algebraic  reasoning  depend  upon  the  presence  of  imaginary 
symbols.  (3.)  An  imaginary  result  generally  indicates,  that  we 
have,  in  some  way,  introduced  inconsistent  conditions  into  our  inves- 
tigation; and  demonstrates  the  impossibility  of  finding,  under  the  cir- 
cumstances, such  a  result  as  we,  at  first,  proposed  to  find. 

POLYNOMIALS. 

v  163.  "We  shall  consider  here  only  the  positive  integral 
powers,  and  simple  roots  of  polynomials.  It  is  evident, 
moreover,  that  if  we  can  find  such  powers  and  roots  of  a 
polynomial,  we  can  find  all  powers.  For  the  formation  of 
the  power  denoted  by  the  numerator,  and  the  extraction  of 
the  root  denoted  by  the  denominator  will  give  any  positive 
fractional  power;  and  the  proper  combination  of  those 
processes  with  division  will  give  all  negative  powers. 

§  164.  The  most  obvious  method  of  finding  a  positive  in- 
tegral power  of  any  quantity  is  by  continued  multiplication 
of  the  quantity  by  itself;  taking  it  as  a  factor  as  many 
times  as  there  are  units  in  the  exponent  of  the  power. 
Thus  we  have  already  found  (§  89) 

(a+x)2  =  (a+x)(a-\-x)  =  a2+2ax-f-x2. 
So  (a-f-x)3  =  {a-\-x){a-\-x){a-\-x)  —  a3+3a2x+ 3ax*+x*. 
(a+x)  *  =  (a+ar)  3  (a+x)  ==  a*+4a3a:-f-6a2a:2+4a2;3-|-a;4. 
(a+x)  5  =  a5+5a*x+10a32;2-f  10a2z3-f 5ax*+ x°. 


132  POWERS  AND  ROOTS.  |J  165-168. 

§  165.  We  find  that,  in  these  instances,  (1.)  the  first  term 
of  each  power  of  the  binomial,  a-\-x,  is  that  power  of  the 
first  term  of  the  binomial;  (2.)  that  the  exponents  of  the 
first  or  leading  quantity,  a,  diminish,  and  those  of  x  increase 
by  unity  in  the  successive  terms ;  (3.)  that  the  exponent  of 
a  in  the  last  term  is  zero,  and  that  of  x  is  the  exponent  of 
the  recpiired  power ;  (4.)  that  the  numerical  coefficient  of 
the  second  term  is  the  same  as  the  exponent  of  the  recpiired 
power ;  and  (5.)  that  the  numerical  coefficients  at  equal  dis- 
tances from  the  two  extremities  of  the  series  are  equal. 

Note.  It  will  be  shown  hereafter,  that  these  principles  apply  to 
all  positive  integral  powers  of  a  binomial,  and  that  all  but  the  third 
and  fifth  apply  to  every  power  of  a  binomial,  whether  the  exponent 
be  positive  or  negative,  integral  or  fractional. 

§166.  "We  have  enunciated  these  principles  as  proved 
only  so  far  as  we  have  found  them  true  by  actual  multipli- 
cation. Let  us  suppose,  that  we  have  found  the  law  of  the 
first  and  second  terms,  given  above  (§  165.  1,  4),  to  be  true 
to  the  nth.  power.     See  §  95.  N.  1. 

Then  we  have     (a-\-x)n  =  an-\-7ian~lx-\-&c. 

Multiplying  by  a-\-x, 

(«4-;r)n+i  —  an+i-\-(n+l)anx-\-&c. 

If  then  the  principles  1,  and  4  of  §  165  are  true  for  the 
nth  power,  they  are  true  for  the  n-\-l  power,  and  so  on, 
without  limit,  n  being  any  positive  integer  whatever. 

§  167.  If  we  substitute,  in  the  above  expressions  (§  164), 
— x  for  -\-x,  we  shall,  evidently,  obtain  the  powers  of  a — x. 
This  substitution  will,  obviously,  cause  all  the  terms  con- 
taining the  odd  powers  of  x  to  become  negative,  and  will 
occasion  no  other  change.     Thus, 

(a—x)  -  —  a2—2ax+x*.     See  §  90. 

(a— xy  =  a3— 3a2a:-f-3aa;2— z3. 

(a—x)*  =  a4— 4a3a:-{-6a5a:2— 4aa;3-fx4. 

$  1 68.    (a-f-z)  2  =  a2-j-2rtcc-Hc9 .     Substitute  b-\-y  for  x 


$  169,  170.]  squart:  root  of  a  polynomial.  133 

Then  («+H-3/)2  =  «2+2«(H^)+(Hiy)2      .  .  (1) 

or  (a+H-y)2  =  («+£) 2+2(«+%-h$/2  •  •  •  (2) 

Developing,  (a+b+y)  2  =  «2+2a5+&2+2ay+2fy+2/2. 

That  is,  The  square  of  the  sum  of  three  members  is  equal 
to  the  sum  of  their  squares,  plus  twice  the  sum  of  their  pro- 
ducts, taken  two  and  two. 

Note.  By  increasing  the  number  of  terms,  we  might  find  similar 
expressions  for  the  square  of  any  polynomial.     Thus, 

(a+b+c+z)  2  =  (a+b+c)2+2(a+b+c)z+z*. 

Hence,  The  square  of  any  polynomial  is  equal  to  the  sum  of  the 
squares  of  the  terms,  plus  twice  the  sum  of  their  products,  taken 
two  and  two. 

§169.  (a+x)3  =  a3+3a2a;+3ax2+a:3.     Substitute 

b-\-y  for  x.     Then 

(«+Hi/)3  =  «3+3«2(H^)+3«(H^)2+(H^)3  •  W 

or(a+6-f-y)3  =  (a+5)3-L.3(a+5)2y+3(«+%241/3  .  (2) 
.-.  (a-L.J-fy)3  =a3-{-3a2b-{-3ab2-\-b3+3a2y-\-Qaby+3bSy 

+3ay  2+3fy2  +#3  =  as+b*+yS+Za*  (b+y)+3b*  (a+y) 

+3y*(a+b)+6aby. 
That  is,  The  cube  of  a  trinomial  is  equal  to  the  sum  of 
the  cubes  of  the  terms,  plus  three  times  the  square  of  each 
term  into  the  sum  of  the  other  two,  plus  six  times  the  product 
of  the  three  terms. 

Notes.  (1.)  We  might  find,  in  like  manner,  expressions  for  the 
higher  powers  of  a  trinomial.  (2.)  If  one  of  the  terms  of  the  tri- 
nomial becomes  zero,  the  formulae  of  §§  168,  169  give  the  square  and 
•ube  of  a  binomial. 


SQUARE  ROOT  OF  A  POLYNOMIAL. 

§  170.  Find  the  square  root  of  a2+2ab-\-b2. 

a.)  The  polynomial  being  arranged  according  to  the  des- 
cending powers  of  a,  we  know,  that  a2  must  be  the  square 
©f  one  term  of  the  root  (§§  73.  1 ;  82.  a,  b). 

b.)   We  know,  moreover,  that  the  polynomial  containa, 

ALG.  12 


134  POWERS  AND  ROOTS,  {^§  171. 

besides  the  square  of  the  first  term,  twice  the  product  of  the 
first  term  by  the  second  (§  168),  and  so  on.  If,  therefore, 
we  divide  the  next  term  of  the  arranged  polynomial  by  2a, 
we  shall  find  another  term  of  the  root. 

c.)  If  now  we  subtract  from  the  given  polynomial  the 
square  of  the  terms  of  the  root  already  found,  the  remain- 
der, if  there  be  one,  will  contain  the  terms  which  resulted 
from  the  multiplication  of  the  remaining  terms  of  the  root 
by  each  other,  and  by  the  terms  already  found  (§  168.  2,  N.). 

d.)  We  may,  therefore,  find  another  term  of  the  root,  by 
dividing  the  first  term  of  the  arranged  remainder  by  twice 
the  first  term  of  the  root ;  and  so  on  (§  82.  c). 


Thus,  a2+2a5+&3 

a* 

2ab+b°- 
2ab+b2 


a+b 
2a+b 


Notes.  (1.)  It  will  be  seen,  that  we  have  subtracted  the  square 
of  the  two  terms  of  the  root  found  (§  170.  c).  For.,  (a+b)^  = 
a^+2ab+b-  =za.2+(2a+b)b.  Now  we  subtracted  a2  at  first,  and 
afterwards  subtracted  (2a+b)b.  (2.)  Also,  after  each  subtraction, 
we  shall  have  subtracted  the  square  of  the  whole  root  then  found 
(§171.  Ex.  1,  a). 

(3.)  As  there  is  no  remainder,  there  can  be  no  other  terms  in  the 
root.  And  whenever  we  find  a  remainder  equal  to  zero,  the  work  it 
completed  (§82.  g),  and  the  given  polynomial  may  be  said  to  be  a 
perfect  power. 

(4.)  If,  however,  after  exhausting  the  given  terms  of  the  polyno- 
mial, we  still  have  a  remainder,  the  root  cannot  be  exactly  found  by 

this  process. 

(5.)  We  may,  however,  continue  the  process,  and  develop  the 
root  in  an  infinite  series,  as  in  division  (§87). 

From  the  reasoning  above,  we  deduce  the  following 

RULE. 

§  171.   1.   Arrange  the  polynomial  according  to  the 
powers  of  some  letter. 

2.   Extract  the  root  of  the  first  term  for  the  first 


§  172.]  SQUARE  ROOT  OV  A  POLYNOMIAL.  loO 

term  of  the  required  root;  and  subtract  its  square 
from  the  given  polynomial. 

3.  Double  the  part  of  the  root  already  found,  for  a 
partial  divisor ;  and  divide  the  first  term  of  the  re- 
mainder by  the  first  term  of  the  doubled  root;  setting 
the  quotient,  with  its  proper  sign,  as  a  term  both  of  the 
root  and  of  the  divisor. 

4.  Multiply  the  divisor  thus  completed  by  the  new 
term  of  the  root,  and  subtract  the  product.  Continue 
the  process  as  long  as  the  case  may  require. 

1.  (Ox*— 12x;3+16a:2— 8x-f-4)^  =  what? 
9z4— 12z3-j-l&c2— 8a>f4  |  3;r2— 2a:+2 
9a:* 

— 12a:3  j  6a:2— 2x 
— 12or34-4a;g 

12a;2  j  6z3— 4x-f-2 

12a;2— 8*4-4 

a.)  We  must  be  careful,  at  each  step,  to  double  the  whole 
of  the  root  already  found,  for  a  divisor.     For 

(a+i+c)2  =z  (a+b)*-\-2{a+b)c+c2.         §  168.  2. 

Also,  (a-f-S+c-fa-)  2  =  {a+b+ c)2+2{a+b+c)x+x2 ;  and 
eo  on.     §  168.  N. 

2.  What  is  the  square  root  of  a*-f-4a3J-f-6a252+4ai'v 
+b*  ?  Am.  a*-\-2ab+b*. 

3.  (16x*+24a;34-89a;2+60a;+100) *  =  what  ? 

4.  (a±  2a?$+b)  *  =  what  ? 

-4»s.  a2±52,  or  <Ja±*/b. 

5.  (a2+2aa:S-|-aO*  =  what?     (x2-|-pa;+i^2)^? 

6.  (a3^fc2a"o:»-f^")^=what?  J/w.  o"±x". 

§  172.  J.)  The  square  root  of  a  trinomial  perfect  power 
may  be  immediately  determined  by  inspection.     For  the 


136  POWERS  AND  ROOTS.  [§  173, 

roots  of  the  terms  containing  the  highest  and  the  lowest 
powers  of  the  letters  being  extracted,  ihe  remaining  term 
must  contain  twice  the  product  of  those  roots  (§  89).  More- 
over, if  this  double  product  of  the  roots  is  positive,  they 
must  have  like  signs ;  if  negative,  unlike.  Hence,  to  ex- 
tract the  root  of  a  trinomial  perfect  power,  extract  the  roots 
of  the  terms  containing  the  highest  and  the  lowest  jjoivers  of 
the  letters,  and  give  them  like  or  unlike  signs,  according  as 
the  remaining  term  is  positive  or  negative.     See  §  93.  I.  II. 

(a2±2ax+x-f  =  a±x. 

(n*±2n-\-l)?  =  what  ?     (64a2-f  I12ab+m2)  ? 

§173.  c.)  1.  Extract  the  square  root  of  a 2+x2.     §170= 
N.  4. 

a2-f-:r2  |  a-\-\a~^x2— Ja~3a:4+&e.  =  a  (1  +  \a~2  x2— 

^a-tx^-j-tScc. 


a2 

x2  |  2a+\a~^x2 
x*-\-\a-2x± 

—\a~2x±  |  2a-\-a~^x2— ±a~3x* 

—la-2x±—\a-'ixS-\-^a-<ix* 

\a~*x& — ^a~ex8 

Here  the  second  term  of  the  root  (§  171.  3)  is  x2-1r2a  — 

a;2 
ia~^x2  =  —  (§  SO.  a).     Thus,  in  another  form, 
2a 

I-  ,  x2  x±         x6  5x6      ,  „ 

v       '       /  2a  8a3       16a5       128a' 

..       x2        x±         x*  5x* 

V  ^2a2      8a4  ~  16a6  128a8  ~       \ 

d.)   Otherwise,  (a2+*2)*  =  (a2)*(l-j-a-stf*)*  =  a  (1 

+ a-2«2)i     See  §§  155,  181. 

Substitute  y  for  a~2x2,  and  extract  the  root  of  1-j-y. 

We  thus  find  (1-H)^=  l+L'/-1si/2+iJtr2/3-T^2/4+&c 

Then  substituting  for  y  its  value,  and  multiplying  by  a, 
we  have  the  same  result  as  before, 


§  174.]       SQUARE  ROOT  OP  NUMBERS.  137 

Let  o  =  10,  and  a:  =  l;  then  (a2+x2)^  =  (101)^=10(1 
+&c.)  =  10(1  +  .005  —.0000125  +&c.)  = 


1  200      80,000 
10(1.0049875+&c.)  =  10.049875  &c. 

2.   (#2— cc2)^  =  what? 

_,,      la;2       Ix*       lx6      „    v 
^.i2(l-__-_____&c.) 

§  174.  In  like  manner,  in  Arithmetic,  we  extract  the 
square  root  of  the  greatest  square  contained  in  the  left  hand 
period ;  subtract  the  square ;  divide  the  remainder  by  twice 
the  part  of  the  root  found ;  set  the  new  figure,  at  the  same 
time,  in  the  root  and  in  the  divisor ;  and  multiply  the  divi- 
sor so  completed  by  the  new  figure  of  the  root. 

1.    Extract  the  square  root  of  5569G. 

5'56'96  |  200+30+6        or        5'56'96  I  236 


4^00  00 

4 

156  96  |  400+30 

43)156 

129  00 

129 

27  96  |  460+6 

466)2796 

27  96 

2796 

a.)  The  terms  are  not  distinct  in  Arithmetic  as  in  Alge- 
bra. But  it  is  evident,  that  the  square  of  the  unit  figure 
must  be  found  in  the  first  and  second  places  on  the  right ; 
the  square  of  the  tens,  in  the  third  and  fourth  places ;  and 
the  square  of  the  hundreds,  in  the  fifth  and  sixth. 

b.)  Hence,  if  we  separate  the  number  into  such  periods 
of  two  figures  each,  the  square  of  the  highest  figure  of  the 
root  will  be  contained  in  the  left  hand  period ;  and,  when 
subtracted,  will  leave  twice  the  product  of  that  figure  by 
the  other  figures,  together  with  the  squares  of  the  other 
figures. 

c.)  Moreover,  the  double  product  of  the  first  and  second 
figures  of  the  root,  together  with  the  square  of  the  second 
figure  will,  obviously,  be  contained  in  what  remains  of  the 
first  two  periods  on  the  left. 

*12 


138  POWERS  AND  ROOTS.  [§  175. 

d.)  Consequently,  we  must  divide  the  remainder  of  the 
first  two  periods  by  twice  the  first  figure  of  the  root,  re- 
garded as  denoting  tens ;  and  add  to  the  partial  divisor  the 
figure  thus  obtained  for  a  complete  divisor. 

e.)  Then  the  remainder  of  the  first  two  periods,  together 
with  the  third,  will,  obviously,  contain  the  double  product 
of  the  two  figures  of  the  root,  already  found,  by  the  third 
figure,  together  with  the  square  of  the  third ;  and  so  on. 

Notes.  (1.)  The  terms  of  the  power  not  being  distinct,  the  dou- 
ble of  the  part  of  the  root  already  found  is  only  a  trial  divisor;  and 
the  correctness  of  the  next  figure  of  the  root  can  be  verified  only  by 
multiplying  it  into  the  complete  divisor,  and  subtracting  the  product. 
(2.)  The  trial  divisor,  on  account  of  the  local  value  of  figures,  forms 
a  large  part  of  the  complete  divisor,  and  is  therefore  an  approxima- 
tion to  it.  (3.)  (a+l)2 — a2=2flrhL.  If,  therefore,  the  remain- 
der is  not  less  than  twice  the  root  found,  plus  one,  the  last  figure  la 
too  small. 

§  175.  f.)  If  after  obtaining  the  last  integral  figure  of 
the  root  we  have  not  a  remainder  equal  to  zero,  the  given 
number  is  not  a  perfect  square ;  and  its  root  cannot  be 
found  but  by  approximation.  For,  if  a  mixed  number 
(§  112)  could  express  the  exact  root  of  a  whole  number,  the 
mixed  number  being  reduced  to  an  improper  fraction 
whose  terms  (§111.  N.)  are  prime  to  each  other,  the  square 
of  this  fraction  must  be  a  whole  number. 

But,  if  two  numbers  are  prime  to  each  other,  the  product 
of  any  number  of  factors  equal  to  the  one  will,  evidently, 
be  prime  to  the  product  of  any  number  of  factors  equal  to 
the  other.  For  such  a  combination  of  prime  factors  can 
never  introduce  a  common  factor.  Consequently  any  pow- 
er whatever  of  the  numerator  will  be  prime  to  the  same 
power  of  the  denominator ;  and  the  square  of  the  improper 
fraction  which  we  supposed  to  be  the  root  of  the  whole 
number,  must  be  an  irreducible  fraction,  and  not  a  whole 
number.  Hence  no  irreducible  fraction  can  be  the  root  of 
a  whole  number  ;  and,  if  the  root  of  a  whole  number  is  not 
a  whole  number,  it  cannot  be  expressed  at  all  except  by  ap- 
proximation (§  153.  N.). 


§176,177.]        CUBE  ROOT  OF  A  POLYNOMIAL.  139 

§  176.  g.)  The  approximation  to  a  surd  (§  153)  root  is 
effected  on  the  same  principle  as  the  simplification  of  a  rad- 
ical fraction  (§  157) ;  i.  e.  by  reducing  the  number  to  a  frac- 
tion, whose  denominator  is  a  perfect  power.     Thus, 

**=(-5i-j  =(25)  =5'  Wlthin  *• 

rt       fti       /2Xl02xi       /200x*      14      ,.      .,.      , 

0r'   2 *=(-ioi-)  -(100) '=io=1^wflnn  ** 

Affun2 -(loor)  =(ipoo)  =  L41'  Wlthm 

.01. 

The  greater  the  denominator,  the  closer,  obviously,  is  the 
approximation.  For,  the  root  of  the  numerator  being  ex- 
tracted to  the  nearest  unit,  the  root  of  the  fraction  is  found 
within  a  unit  divided  by  the  root  of  the  denominator. 

h.)  The  approximation  is,  of  course,  most  conveniently 
performed  with  the  powers  of  10, 100, 1000,  &c.  And  this 
is  the  ordinary  process  of  approximation  in  Arithmetic,  in 
which  the  denominator  is  not  written  ;  and  the  approxima- 
tion may  be  carried  to  any  extent,  by  annexing  new  peri- 
ods of  cyphers  to  the  number  (i.  e.  by  multiplying  it  repeat- 
edly by  10 2),  and  thus  extending  the  root  to  additional 
places  of  decimals  (i.  e.  dividing  the  root  repeatedly  by  10). 

i.)  In  like  manner,  if  the  terms  of  a  vulgar  fraction  are 
not  perfect  powers,  we  can  generally  extract  its  root  most 
conveniently,  by  first  reducing  it  to  a  decimal.  If  it  redu- 
ces to  a  repeating  decimal,  instead  of  annexing  cyphers  in 
approximating  we  should,  of  course,  annex  figures  of  the 
repetend. 

CUBE  ROOT  OF  A  POLYNOMIAL. 

§  177.  Find  the  cube  root  of  a*-\-3a*b-\-§ab*+b3. 

a.)  Reasoning  as  in  respect  to  the  square  root  (§  170.  a), 
we  arrange  the  polynomial,  extract  the  cube  root  of  the 
first  term,  and  subtract  the  cube. 


140  POWERS  AND  BOOTS.  [§  178. 

b.)  We  then  know,  that  the  first  term  of  the  arranged 
remainder  will  consist  of  three  times  the  square  of  the  first 
term  of  the  root  into  another  term.  We  may,  therefore, 
find  another  term  of  the  root  by  dividing  the  first  term  of 
the  remainder  by  three  times  the  square  of  the  first  term 
of  the  root.     See  §  169. 

c.)  If  now  we  subtract  from  the  given  polynomial  the 
cube  of  the  part  of  the  root  already  found,  the  first  term  of 
the  arranged  remainder,  if  there  be  one,  will  contain  three 
times  the  square  of  the  first  term  of  the  root  into  another 
term  (§  1G9) ;  and  so  on. 

d.)  The  cube  of  a-\-b  consists,  besides  a3  already  sub- 
tracted, of  3a2b+3ab2+b3  =  {3a*+3ab+b°-)b.  The  most 
convenient  method,  therefore,  of  completing  the  subtraction 
of  (a-\-b)3,  is,  after  having  found  b  by  dividing  the  first 
term  of  the  remainder  by  3a2,  to  form  the  polynomial  fac- 
tor 3a'2-\-3ab-\-b'2,  and  then  multiply  it  by  b.  That  is,  we 
may  add  to  three  times  the  square  of  the  first  term,  three 
times  the  product  of  the  two  terms,  and  the  square  of  the 
new  term ;  and  multiply  the  sum  by  the  new  term. 


Thus,      a3-}-3aH-\-3ab2-\-b3 


a 


3 


3a2b+3ab2+b3 
3«2&-f-3aJ2-f-53 


a-\-b,  Hoot. 
3a2+3a5-}-5a,  Divisor. 


Notes.  (1.)  We  have,  evidently,  subtracted  the  cube  of  the  two 
terms  of  the  root.  For,  (orB)3  =  a3+Ba^b+Bab^+b3=za3+(Za2 
+3ab+b*)b.  (2.)  Remarks  similar  to  §  170.  N.  2,  3  4,  5  apply 
equally  to  the  cube  root.  But  the  approximation  by  this  process  to 
the  cube  roots  of  imperfect  powers  is  so  laborious,  that  other  meth- 
ods, which  will  be  considered  hereafter,  are  preferable. 
From  the  reasoning  above  we  have  the  following 


RULE. 

§  178.  1.  Arrange  the  polynomial  according  to  th 
powers  of  some  letter. 


<!  179.J  CUBE  ROOT  OF  A  POLYNOMIAL.  245 

2.  Extract  the  cube  root  of  the  first  term,  for  the 
first  term  of  the  root,  and  subtract  its  cube. 

3.  Divide  the  first  term  of  the  arranged  remainder 
by  three  times  the  square  of  the  first  term  of  the  root. 

4.  Add  to  three  times  the  square  of  the  part  of  the 
root  previously  found,  three  times  the  product  of  the 
previous  part  of  the  root  by  the  new  term,  and  also 
the  square  of  the  new  term  (§  177.  d). 

5.  Multiply  the  divisor  so  completed  by  the  new 
term  of  the  root;  subtract,  multiply  the  square  of  the 
whole  root  already  found  by  3,  divide,  complete  the  di- 
visor, multiply  and  subtract;  and  continue  the  process 
as  long-  as  the  case  may  require. 

1.     (te—6t5+l5t±—20ts-\-15t*—6t+l)^  —  what  ? 


t* 

— 6t5+l5t*— 20t3+l5t2— 6*+l 

— 6*5-j-12^—  &3 

3t*— 12t3+lot2— 6*+l 
3**— 12t3-\-15t2—6t+l 


t*—2t-\-l,  root. 

3t*— 6t3+it2,  1st 
divisor. 
3t*—12t3+l5t*—G( 
-\-l,  2d  divisor, 


2.  (aG+3a*x2+3a2x*-\-xrf  =  what? 

3.  (JTx3+§z2+4x+8)^==what?  Ans.  \x-\-2. 

4.  (a*±3,a-\-%a?±%y3=xvhat?  Ans.  o?±\. 

§  179.  In  like  manner,  in  Arithmetic,  the  number  being 
separated  into  periods  of  three  figures  each,  (because  the 
cube  of  the  unit  figure  must,  evidently,  be  found  in  the  first 
three  places,  the  cube  of  the  tens  in  the  next  three,  and  so 
on,)  we  extract  the  root  of  the  gi-eatest  perfect  cube  in  the 
left  hand  period ;  subtract  the  cube  from  that  period ;  di- 
vide the  remainder  of  that  period,  with  the  next,  by  three 
times  the  square  of  the  first  figure  of  the  root  regarded  as 
standing  in  the  place  of  tens ;  then  complete  the  divisor, 
multiply,  subtract,  bring  down  the  next  period ;  and  divide 


1  i2  POWERS  AND  ROOTS.  [§  180. 

by  three  times  the  square  of  the  whole  root  already  found, 
regarded  as  denoting  tens,  and  so  on. 

What  is  the  cube  root  of  1953125  ? 
1'953'125  J  125 
1 

953  |  300+60-1-4  =  364,  1st  complete  divisor. 

728 

225125  1  43200+1800+25  =  45025,  2d  complete  divisor. 

225125 

Notes.  (1.)  The  terms  of  the  power  not  being  distinct,  three 
times  the  square  of  the  part  of  the  root  already  found  is  only  a  trial 
or  approximate  (§  174.  N.  2)  divisor,  and  the  correctness  of  the  next 
figure  of  the  root  can  be  verified  only  by  multiplying  it  into  the  com- 
pleted divisor,  and  subtracting  the  product.  (2.)  (a+1)3 — a3=  = 
3a2+3a+l.  If,  therefore,  the  remainder  is  not  less  than  three  timei 
the  square  of  the  root  found,  plus  three  times  the  root,  plus  one,  the 
last  figure  is  too  small. 

WW  ROOT  OF  A  POLYNOMIAL. 

§  180.  We  know  that  (a-\-b)n=:  a"+«an-15+&c.  (§  166). 
Hence  we  have,  for  finding  the  nth  root  of  a  polynomial, 
the  following 

RULE. 

1.  Arrange  the  polynomial,  extract  the  nth  root  of  the  first 
term  for  the  first  term  of  the  root,  and  subtract  its  power. 

2.  Divide  the  first  term  of  the  arranged  remainder  by  n 
rimes  the  (n — 1)  power  of  the  first  term  of  the  root. 

3.  Raise  the  whole  root  so  found  to  the  n01  power  and 
subtract  it. 

4.  Divide  the  first  term  of  the  arranged  remainder  by  the 
same  divisor  as  before,  subtract  the  nih  power  of  the  whole 
root  from  the  given  polynomial,  and  so  on. 

a.)  If  we  make  n  =  2,  we  have  a  rule  for  the  square 
root ;  if  n  ~  3,  for  the  cube  root. 


$  181,  182.]     n<h  root  of  a  polynomial.  143 


1.    (a5— l0a4x-{-40a3x2— 80a2x3H-80ax4— 32x •"')' 
what? 
a  5_i0a4a._i_40a3a.2_30a2a.3_L.80ax4— 32a;5  |  a— 2x 


-10a4x  I  5a4,  divisor. 


a  s— 10a4x+40a3x2— 80a*x3-\-80ax*— 32x5. 

2.    (I6a4+96a3x+216a2x2+216ax3+81x4)^_what? 

Ans.  2a-f-3x. 

b.)  In  the  last  example,  the  root  may  be  more  easily 
found  by  extracting  the  square  root  twice.  And,  in  gene- 
ral, whenever  the  number  of  the  root  is  a  product  of  two  or 
more  numbers,  we  may  extract  successively  the  roots  indi- 
cated by  the  several  numbers. 

Thus,  to  find  the  sixth  root,  we  may  extract  the  square 
root,  and  then  the  cube  root ;  to  find  the  eighth  root,  we 
may  extract  the  square  root  three  times ;  and  so  on. 

c.)  It  is  best  in  such  cases,  if  the  roots  are  of  different 
decrees,  as  the  square  and  cube  roots,  to  extract  the  lowest 
root  first. 

§  181.  There  is  frequently  an  advantage  in  simplifying 
($  155)  the  expression  of  a  root  of  a  binomial,  or  of  any 
polynomial  which  is  not  a  perfect  power.  See  §  173.  d. 
Thus, 

(a3— a2x)2_  (a2)^(a— x)2_a(a— x)i 

(«3_L.2a2a;-j-ax2)^--z  (a2+2ax-{-x2)%2  _  (a-\-x)a*. 

(a*— o3s;2)^  —  a(a2— x2)^. 

SQUARE  ROOT  OF  CL±b~ . 

§  182.  The  square  root  of  a  binomial  of  the  form  a±bJ 
can  sometimes  be  obtained  by  a  peculiar  process,  which 
depends  on  the  following  principles. 


144  POWERS  AND  BOOTS.  [§  183,  184. 

§  183.  I.  Let  a  and  x  be  rational,  and  «Jb  and  «/y,  irra- 
tional;  then  if  a±^/b  =  x±*/y,  a  will  be  equal  to  x,  and 
Jb  to  Jy. 

For  t  equal  to  x,  let  it  be  equal  to  x±c. 

i         i 

Then         x±c±.+/b  —  x±.«/y;  or  c±5'J  =y*. 

.-.  Squari    >  c2±2c52-|-5=ry. 

X  y — c2 b 

b2  =  ±£ — (§  42.  o,  c?) ;   that  is,  an 

irrational,  equal  to  a  rational  quantity,  which  is  absurd 
($  175).     See  Geom.  §  23.     Hence, 

Two  binomials,  consisting  each  of  a  rational  and  of  an 
irrational  term,  cannot  be  equal,  unless  the  rational  terms 
are  equal  to  each  other,  and  also  the  irrational. 

§  184.    Let  (a-\-b*y=.x *-\-y  ,  one  or  both  of  the  quan- 

x  i 

tities  x'J  and  y'2  being  irrational,  and  x  and  y  monomial. 

Then,  squaring, 

a+5*  =  x+2x2y2-\-y ;  or  a+^b  =  x+2^(xy)+y. 

a  =  x+y,  and  b2  =  2x^y2  (§  183). 
Hence,  subtracting, 

a— b2  =  x—2x2y^-\-y  =  (x^—y^)  2. 

(a— b2)^  =  x^—y2  (§  52.  N.).     That  is, 
If  +/{a-\-jV)  zrz^x-^y/y,  then  +/(a — +/b)  = +/x — +/y. 

Thus,         (3+52)2  =  9+6x5^+5  =  14+6X5  2"; 

and  (3— 52f  =  9— Gx5*+5  =  14—6x5^. 

.-.       y (14+6^5)  =  3+^5,  and  y(14— 6^/5)  =  3—^/5. 

(2^±32)  2  =  2±2x2i3*+3  =  5*2x2^.3^. 
^/(5±2y  (2X3))  =  y2±y3. 

Note,  x  and  y  being  monomials,  the  squares  of  Jx  and^/y 
must  be  rational,  and  will,  of  course,  combine  by  addition,  into  a 
tingle  rational  term  a;   while  their  double  product,  being  equal  to 


5  ^-J  SQUARE  ROOT  OF  a±b*. 

/b  (§  181),  is  irrational,  and  will  be  positive  or  negative  according 
as  Jx  and  Jy  have  the  same  or  different  signs. 

§  185.   Now  assume  (a-\-$)%  =  x^+y^  (1) ; 

then  (<*-&*)*=  x^-y^  (2).  §  184. 

Squaring  (1)  and  (2), 

a-\-b^  —  x-\-2x^y2-\-y,  and 

a— $=x— 2x^-\-y. 

Adding,  and  dividing  by  2,  we  have 

a  =  x+y{3). 
Again,  multiplying  together  (1)  and  (2),  we  have 

(a2— b)^  =  x—y{4).*  §92. 

Hence,  from  (3)  and  (4), 

,.  ti= (fi±±V)\  and  ,i= (fc^zS*)* 

Hence,  substituting  in  (1)  and  (2), 

(^i=(f±rf)i+(f=(^)4 

Or,  putting  (a2 — b)2  =  c,  we  have 

(a+Jl)l=(2f)i+(2=?)* 

or  y  (o+y*)  =  y-f-  +^^-  : 


(a+62)i(a-6l)l  =   [(a+&2-)(a-&2)]£    [§  151.    a] 

-6)2  [§92]. 

ALG.  IS 


146  POWERS  AND  ROOTS.  [§  I 

(«-h-)--\rY)  -vt;  ' 

a-\-c         a — c 
or  */(«— x/o)  =  y— ^~2~ ' 

Note.  These  expressions  will,  evidentty,  not  reduce  to  a  con- 
venient form,  unless  (a2 — b)'$  is  rational,  i.  e.  unless  a2 — b  is  a 
perfect  square. 

The  above  results  may  be  verified  by  squaring.     Thus, 

Jf-C^  =  «±(«2— c2)2"  =  «±(a3— (a2— J))^  =  «±^. 

1.    (3-f-2.y2)*  =  what? 

Here  a  =  3,  and  &*  =  2(2)*=  (22.2)*  =  8*  (§  161). 
c  =  (a2— ft)*  =  (9— 8)*  =  1*  =  1. 

••■  C4c)i+(?)i=(^i)i+(2ii)i=2-+1- 

Wc  may  verify  this  result  by  squaring  22-j-l.     Thus 
(2M-1)2  =  2+2(2)^4-1  =  3+2^2. 

(9±4X5*)*  =  what  ?  Ans.  2±oK 

3.    (7±2XlO-)-=what?  Ans.  5-±2J. 

(£±(f)"*)*  =  what?  Ans.^±<J)K 

^(6-1-6^— 3)=  what? 

Here  «  =  G,and  6*=6(-3)*  =  (62(-3))*  =  (-108)*. 

...  &- -108,  and  c=(a3—5f  =(36— (—108))*  =  (144)* 
=  12. 

(6+G(-3)^)2  =  3-h(-3)2. 

6.  y(2+V— 2)+y(2— 4y— 2)  =  what?      -4«s.  4. 

7.  \bc+2b(bc— ft*)*]*  +  [bc-n{bc-b"-yf  -  ±26. 


§186,187.]  BINOMIAL  SURDS.  147 

8.    [ab+lc2— tf 2+2(4aJc2— abd°~) *]*  =  what  ? 

Ans.  Jab-\-*/±c- — d-. 

§  186.   Such  expressions  as   a±*/b,  or  ^/a±^/b,  are 
sometimes  called  binomial  surds. 

We  have     (a+&*) («— &"2)  =  a2— b  (§92). 

Also  («^±J^)(a*q:i-)  =  a— 6.     Hence, 

J%e  product  of  the  sum  and  difference  of  two 

roots,  or  of  a  square  root  and  a  rationed  quantity  iri- 

rational. 
Thus,         (2-K/5)  (2— yo)  =  4—5  =  — 1. 

§  187.    a).   This  principle  is  frequently  useful  in  fie 

>ne  of  the  terms  of  a  fraction,  or  one  of  the  members  of  an 

equation,  of  irrational  expressions.    Thus,  let  it  be  required 

2 73 

to  reduce  j— j- — -  to  a  fraction  having  a  rational  denomina- 

tor. 

We  have     (W3)X  ^~^  -  <t?W  -  7-4^3 
We  have     (2+v3) x (2_y 3)  -     4_3      -<     *S*- 

1.  Reduce  in  like  manner  — — — — -.     Ans.  ,x/8:f.v/3, 

2.  Reduce  in  like  manner  — ^  ,    . 7- ^. 

y(i+^)-y(i-x) 

i+y(i— 


Ans 


y(a»+l)-l\i, 

'     V^02_|_l)_L.l 

Rendering  the  denominator  rational  by  multiplying  both 
terms  by  the  numerator,  there  results 

{ [V(*3+i)-ip  I  *_  y02-K)-i 


4.    Simplify  the  fraction 


x 
ax 


y(«2+x2)+a: 


Ans.  -Ua2-\-x-Y— a  '. 


148  POWERS  AND  BOOTS.  [§188-190. 

5.    Given  — =  Jx-\-7,  to  find  x.      Arts.  cc  =  59. 

b.)  If  the  expression  consist  of  more  than  two  terms,  we 
may  proceed  as  follows  : 

(7  ^+5^-f-3*)  (7  24-5^—3^)  =  9+2x7*5^. 

(9-1-2x7^5*)  (9— 2"X7^.5*)=81— 47.5=81— 140=— 59 

§  188.  c.)  If,  instead  of  the  square  root,  one  or  both  of 
the  terms  of  the  binomial  consist  of  higher  roots,  whose 
numbers  are  powers  of  2,  a  repetition  of  the  process  will 
result  in  a  rational  expression.     Thus, 

(a*+&^)  (a*— $)  =  a^—  b*  ;  {a^—b*)  (a?+$)  =  a—$ ; 

i  i 

and  (a — b-)  (a-\-b-)  =  a — b, 

n  n  1  i 

§189.   d.)  We  have  (§96.  d)  a"—  6==  (a"")"  — (b"f  = 
1     i. 
a — b  divisible  by  a"  — bn . 

Dividing,  as  in  §  96.  a,  we  have 

1       1  "— 1       ^— 2  1  1  *—  2       m— j; 

(a— J)-r-(a"— 5")  =  a  "    -fa  "   £+  .    .+a^5~"-f-&  "    . 

j  j  n — 1  « — 2     j  -    n — 2  »i — 1 

[oF—fr)  (a~7r"+a~"~  6""-f- .  +«"&""""+&"""")  =  a— 5. 
Thus,  («M)(a^M)  =  «-5. 
§  190.   e.)   Again  (§  97),  a2"— £2"  is  divisible  by  «+6  ; 
hence  a — b  is  divisible  by  ain-\-b *n.     Dividing,  we  have 

1  j  2«— 1'  2«— 2       .  2n— I 

2n— 1  „2n— 2       ^  2w— 1 

.-.  (a*Hrft*)<«  2"    — «  2'1    *    +  •  •  •  •   — h  2n    =«— & 

Thus,  (a*+&*)(a*— cftfi+ah11— a*ft*+a*ft*— b%)  = 
a— J. 

Also.  (5*+3*) (5^-5^3^+5*3^— 3*)  =  5—3  =  2, 


>  191-194.]  EQUATIONS   OF  THE  SECOND  DEGREE. 


§  191.  /.)  Again  (§  98),  a+b  is  divisible  by  «2'i+1-[-//" 
Thus, 

1  I  2n  2»—  1  ] 

(a+o)-T-(a         +6         )=a         — a         o         4~  .  .  . 

1  on — 1 


2«+l  7  2-1+1     ,     7 
12"  2 " 


(«         -j-6         )  (a         — -\-b         )  —  a-\-b. 

Thus,  {J+$)(a?— a^4-#)  =  «4-o- 
So,  (7*4-4*)  (7^— 7*4*4-4?)  =  7-}-4  =  11. 


CHAPTER  VII. 


EQUATIONS  OF  THE  SECOND  DEGREE. 


S  192.   We  shall,  at  present,  consider  only  equatio 
which  the  exponents  of  the  unknown  quantities  art 
-  gral. 

With  this  limitation,  an  equation  is  of  the  se< 
degree,  when  the  difference  between  the  highest  a 
the  lowest  degrees  of  its  terms  with  respect  to  \ 
known  quantity  or  quantities  (§28.  b)  is  two  (§40. 

§  193.    An  equation  containing  but  one  unknoivn 
quantity  is,  therefore,  of  the  second  degree,  when 
difference  between  the  greatest  and  least  exponent 
the  unknown  quantity  is  two. 

§194.   Notes.     (1.)    We  shall,  at  present,  confine  our  atl 
to  equations  containing  but  one  unknown  quantity;  and  shall  sup  c 

•18 


150  EQUATIONS  OF  THE  SECOND  DEGREE.  [§  195-197, 

them  to  be  arranged  according  to  its  descending  powers  (§  33),  and 
to  be  reduced  to  the  simplest  form  in  respect  to  each  of  those  pow- 
ers (§34.  c).  (2.)  Then,  each  power  of  the  unknown  quantity,  to- 
gether with  its  coefficient  (whether  monomial  or  polynomial),  will 
constitute  a  term  of  the  equation.     Thus, 

Let  x  2+2ax+6  2 — mx  2 — 4x  =  7ix+r — q — 3x  2. 

Then  (1+3— m)x2+(2a— 4— n)x+bz+q—r  =  0.  §§  33,  34.  c,  44. 
Or,  making  .#  =  1+3—  m,  B  =  2a— 4— n,  and  C  =  b2+q— r, 

Ax^+Bx+C~=0. 

§  195.  An  equation  of  the  second  degree,  containing  but 
one  unknown  quantity,  its  powers  being  all  integral,  may 
contain  any  three  consecutive  powers,  and  no  more. 

For,  if  there  were  more  than  three  consecutive  powers, 
or  if  there  were  three  powers  not  consecutive,  the  differ- 
ence between  the  greatest  and  least  exponent  must  be  mor 
than  two. 

Thus,    Ax3-{-Bx*-\-Cx  =  0,  Axn--{-Bx+C=0, 
Ax+B+  Gc-1  (  =  Ax+Bx°+Cxr  *)  =  0, 
and  Ax~r-\-Bx---\-  Gx~3  —  0  are  all  of  the  second  degree. 

§  196.  Hence  an  equation  of  the  second  degree,  when 
reduced  as  above  (§  194),  can  consist  of  only  three  terms 
(§  194.  2) ;  anil  therefore,  an  equation  of  the  second  degree. 
consisting  of  three  terms,  is  called  a  complete  equation. 

§197.   Let  Ax3-\-Bx*+Cx  =  0. 

Dividing  by  x,  Ax*-\-Be+  G—  0. 

Again  let  Ax+B+  Cher »  =  0. 

Dividing  by  x~ y ,  or  multiplying  by  x, 

Ax*+Bx-\-C=0. 

Or,  again,  let     Aar+Ba*- 1  -f  Car-2  =  0. 
Dividing  by  a;"-2,  Axs-\-Bx-\-C=  0.     Hence, 

Every  complete  equation  of  the  second  degree,  containing 
only  one  unknown  quantity,  can  be  reduced  to  the  form 

Ax°-+Bx-\-C=0, 
in  which  the  coefficients,  A,  B  and  G,  may  be  either  post- 


$198,199.]  INCOMPLETE  EQUATIONS.  151 

five  or  negative,  integral  or  fractional,  numerical  or  alge- 
braical, monomial  or  polynomial. 

Reduce  the  following  equations  to  the  above  form. 

1.  ax2-{- bx-\-c—  (mx2-\-nx— p)  =  5x2— 8#-|-7. 

2.  a2-f-2ar  cos  i>-fr2  cos2r+&2+25r  sin  v+r3  sin-  u  = 
i2'3  ;  r  being  the  unknown  quantity. 

Sx— 3  __  9.3x— 6 
a? — o  / 

§  198.  As  the  coefficients  may  have  any  value  whatever, 
they  may  be  equal  to  zero.  But  if  the  coefficient  of  a  term 
becomes  zero,  the  term  itself  becomes  zero,  and  disappears 
from  the  equation.  The  equation  is  then  sometimes  called 
incomplete. 

Notes.  (1.)  If  all  the  coefficients  become  zero  at  once,  the  equa- 
tion will,  of  course,  disappear.  Also,  if  A  and  B  become  zero,  we 
shall  have  C=0,  and  the  equation  will  be  annihilated.  But,  if  A 
and  C  become  zero,  we  shall  have  Bx  =  0,  and  x  =  0.  Again,  if 
B  and  C  become  zero,  we  shall  have  Ax%  r=:  0,  and  x  =  ±0. 

(2. )  Again,  let  A  =  0.  Then  we  shall  have  Bx+C  =  0.  Now 
this  is  no  longer  of  the  second  degree.  It  is  of  the  first  degree,  and 
must  be  treated  accordingly  (§4S).  Neither  of  the  above  supposi- 
tions needs  any  further  consideration. 

§199.   Now  let  B  —  Q.     Then  the  equation  becomes 
Ax2+C=0. 

C  C\ 

x2  —  —  j-=q2  (putting  q2=  —  —  J. 

x=(—^)*=.(g')*=±q.     See  §52.  N. 

In  this  case,  we  find  the  values  of  the  unknown  quantity 
by  reducing  the  equation  to  the  form  x2  =  q~,  and  extract- 
ing the  square  root  of  both  sides. 

Thus,  let  x2  —  49. 

Then  x  =  ^49  =  ±7. 

Note.  The  term,  incomplete  equations  of  the  second  degree,  i* 
sometimes  applied  exclusively  to  equations  of  this  form.  They  are 
also  sometimes  styled  pure  equations  of  the  second  degree,  or  pure 
quadratics  (§41.  N.). 


152  KQUATIONS  OF  THE  SECOND  DEGREE.         [§  200. 

a.)   This  form  of  equation  will,  evidently  have  two  roo 
('  39)  numerically  the  same,  hut  with  opposite  signs  (>  23. 
/•  1). 

b.)   Also,  if  xn-  =  q-,  then  x-—q*  =  0. 

(x+q)  (x—q)  =  0.  §  93.  111. 

Now  it  is  evident,  that  a  product  will  become  zero,  only 
when  one  of  its  factors  is  zero.     The  last  equation,  there- 
fore, will  be  true,  when  either  of  its  factors  is  equal  to  zero 
1  in  no  other  case.     Hence  we  may  have,  either 
oc-\-q  =  0,  or  x — q  =  0  ; 
and,  in  either  case,  we  shall  have  the  product 

(x+q)(x— q)  —  x2—q*  =  0. 
But,  if  x~\-q  —  0,  then  x  =  — q, 

I  if  x — q  =  0,  then  x  =  -\-q. 

So  a;2— 49  —  0  gives  (x-\-7)(x— 7)  =  0. 

Whence, 

#-J-7  =  0,  and  a?  =  — 7  ;  or  a; — 7  =  0,  and  a;  =  +7. 

Either  of  these  values  of  a;  will  satisfy  the  equation,  and 
is  consequently  a  root  of  the  equation  (§  39). 

c.)   If  the  equation,  x-  r=  q2,  or  x- — q-  =  0,  be  pu<  un- 
der the  complete  form,  thus, 

x2+0a:— «72  =  0, 
we  shall  have  -\-q — q  =  0,  the  coefficient  of  x '  ; 

and  (~H?)( — ?)  — — l"i  tue  coefficient  of 

So,  in  the  equation,  a?2-|-0a; — 49  —  0,  Ave  have 
+7-7  =  0;  (+7)(-7)  =  -49. 

.  200.  tZ.)  We  find  here  certain  results,  which  will  here- 
ter  be  shown  to  hold  of  all  equations  of  the  second  de- 
■ee,  when  placed  under  the  form,  x2±2px±q2  =  0,  viz. 

1.   The  equation  can  be  resolved  into  two  binomial 

s;  of  which  the  first  term  of  each  is  the  unknown  quan- 
tity, and  the  second  term,  with  its  sign  changed,  is  a  root 

the  equation. 


§  201,  202.J  INCOMPLETE  EQUATIONS.  —  PROBLEMS.     l5u 

2.  The  equation  has  two  roots. 

3.  The  algebraic  sum  of  the  roots,  with  their  sign? 
changed,  is  equal  to  the  coefficient  of  x1. 

4.  The  product  of  the  roots  is  equal  to  the  coefficient  of 
x°. 

Note.  The  student  should  illustrate  and  test  these  principles  by 
applying  them  to  the  roots  of  every  equation  which  he  solves. 

§  201.  e.)  If  the  equation  be  of  the  form  x2-{-q2  =  0,  we 
shall  have 
x2  = — q2,  and,  consequently,  x  =■  ±^/ — q2  =  ±q+/ — 1, 
an  imaginary  result  (§§  23./.  2,  158). 
Thus,  let  z2_|_49  —  q. 

Then        x2  =  —49  ;  .-.  x  =  y— 49  =  ±7^—1. 

Notes.  (1).  These  expressions  do  not  indeed  represent  any  ac- 
tual value,  but  they  are  called  roots  of  the  equation,  because,  when 
substituted  for  x,  they  satisfy  the  equation  (§39).  (2.)  This  imag- 
inary result  indicates  an  absurdity  in  the  conditions  of  the  problem. 
It  is  here  proposed  to  find  a  number,  whose  square  added  to  another 
square  shall  be  equal  to  zero.  That  is,  the  sum  of  two  positive  (§11. 
N.  2)  quantities  is  required  to  be  zero,  which  is,  evidently,  impossi- 
ble.    See  §  162.  N.  3. 

/.)   The  results,  xz=z-\-q^/ — 1,  and  x  —  —qj — 1  give 

x— qj— 1  =  0,  and  x+qj—l  =  0;       §  199.  b. 

and  .-.      (x— qj— 1) (x-\-qJ— 1)  =  x2-\-q2  =  0.  §  200. 

So         (a:— 7  y— 1 )  (x+7y— 1 )  =  x  2+49  =  0. 

§202.    1.    Given  5(x2— 12)  =  (a:2+4),  to  find  x. 

Ans.  x=-±A, 

.     _,.        a;2— 50    ,  x2— 25    .  „    , 

2.  Given  — \-x  = \-x,  to  find  x. 

z  o 

Ans.  x=.  ±10. 

3.  In  a  right  angled  triangle,  the  square  of  the  hypot- 
enuse, or  side  opposite  the  right  angle,  is  equal  to  the  sum 
of  the  squares  of  the  other  two  sides  (Geom.  §  188).  If 
then  the  base  is  4  feet,  and  the  perpendicular  3  feet,  what 
is  the  hypotenuse  ? 

Let  x  =  the  hypotenuse.     Then  x2  =  32+42,  &c. 


154  EQUATIONS  OP  THE  SECOND  DEGREE.         [§  202. 

4.  A  rope  50  feet  long  is  extended  from  the  top  of  a  flag 
staff  40  feet  high,  in  a  straight  line  to  the  ground  on  the 
east  of  the  flag  staff,  and  on  a  level  with  its  foot.  How  far 
from  the  foot  of  the  staff  will  it  strike  the  ground  ? 

Ans.  ±30  feet  (§5). 
.">.    How  far,  if  the  rope  be  45  feet  long? 

Ans.  ±20.615  &c.  feet. 

6.  How  far,  if  the  rope  be  40  feet  long? 

Ans.  ±0  (i.  e.  it  will  strike  the  ground  a1 
the  foot  of  the  staff). 

7.  How  far,  if  the  rope  be  32  feet  long? 

Ans.  ±y— 576  =  ±24y— 1. 

In  this  case,  the  rope, evidently,  will  not  reach  the  ground;  so  thai 
there  is  manifest  absurdity  in  inquiring  how  far  from  the  foot  of  the 
«taft"  it  will  strike  the  ground.  This  absurdity  is  indicated  by  the  im- 
aginary result  (§201.  N.  2). 

8.  Let  the  perpendicular  drawn  from  any  point  of  the 
circumference  of  a  circle  to  the  horizontal  diameter  be 
represented  by  y  ;  and  let  the  distance  from  the  foot  of  the 
perpendicular  to  tbe  centre,  measured  on  the  horizontal 
diameter,  be  denoted  by  x ;  and  the  radius  of  the  circle,  by 
B.  Then  we  shall  have,  for  every  point  of  the  circumfer- 
ence, x2-\-y2  =  R2 ;  ovy2=R2—x2.     §202.3. 

Or,  if  the  radius  be  10  feet,  we  shall  have  R-  =  100,  and 

y*  =  100— a;2. 

What  now  is  the  length  of  y,  when  x  =  0  ? 

Ans.  y=z-\-  10,  or  —10  (§  5). 

9.  "What  is  the  length  of  y,  when  x  =  ±l,  2,  3,  4,  5,  6. 

7,  8,  9,  10  ? 

10.  What  is  the  length  of  y,  when  x  =  ±11  ? 

.  his.  y  =  */ — 21. 
In  this  case,  the  distance  measured  on  the  horizontal  diameter 
from  the  centre,  being  greater  than  the  radius,  extends  beyond  the 
circumference;  and,  of  course,  no  perpendicular  to  that  line  at  it» 
extremity  can  meet  the  circumference.  Hence  the  imaginary  result, 
indicating  an  absurdity  (§201.  N.  2). 


203.]  INCOMPLETE  EQUATIONS.  L55 

9.  It  is  required  to  lay  out  10  acres  of  land  in  a  square. 
What  must  be  the  length  of  one  side  ? 

10.  The  product  of  two  numbers  is  P,  and  the  quotient 
of  the  greater  by  the  less  is  Q.     What  are  the  numbers  ? 

P 

Let  x  zr  the  greater ;  then  —  =  the  less  ;  &c. 
°  x 

03.   Again,  resuming  the  complete  equation, 
Ax2-{-Bx+0=Q, 
if  we  suppose  C=0,  we  shall  have 

Ax^-\-Bx  =  0. 
Dividing  by  x  (§  51),  we  have  an  equation  of  the  first  de- 
gree (§51.  b), 

7) 

Ax-\-B  =.  0  ;  and  .\  x  — — . 

a.)  If,  however,  we  divide  by  A,  we  shall  have 

7?  B  \ 

x2-\-  —x  =  0,  or  x2-{-2px  =■  0  (putting  2p  z=z  —  J . 

Separating  the  last  expression  into  factors,  we  have 
x(x-\-2p)'==:Q ; 
an  equation,  which  will  be  satisfied,  either  when  x=z0,  or 
when  x-\-2p  =  0;  i.  e.  when  x=-0,  or  when  x=z — 2p. 

The  roots,  therefore,  of  this  equation  regarded  as  of  the 
second  degree,  are  0,  and  — 2p  (§  200.  2). 

b.)  In  this  case  also,  the  sum  of  the  roots  with  their 
signs  changed  is  equal  to  the  coefficient  of  x1,  and  their 
product,  to  the  coefficient  of  a;0  (§  200.  3,  4).  The  two  bi- 
nomial factors  (§  200.  1)  are  x — 0  and  x-\-2p. 

Note.  This  form  of  equa/ion  is  frequently  classed  and  solved  a* 
a  complete  equation  of  the  second  degree  (§  206). 

1.  Given  2Rx — x2  =  0,  to  find  x. 

.Ins.  xz=0,  or  2R. 

2.  Given  r- — 2B  cos  v  r  =  0,  to  find  r. 

Ans.  r  =  0,  or  2R  cos  '.'. 

3.  Given  x- — lOx  =.  0,  to  find  x. 


156         EQUATIONS  OF  THE  SECOND  DEGREE.  [§  204,  205. 

§  204.    Returning  now  to  the  complete  equation, 

JB        C 
and  dividing  by  A,  we  have  x2-\~  —x-\-  —  —  0 ; 

B  C 

or,  putting  2p  =  — ,  and  q  2  =  — , 

x*-\-2px-\-q2  =  0. 

a.)  This  is  a  complete  equation  of  the  second  degree 
(§196);  and  it  is  perfectly  general,  since  every  complete 
equation  can  be  reduced  to  this  form  by  dividing  by  the 
coefficient  of  a:2,  and  substituting  convenient  symbols  for 
the  coefficients  of  a;1,  and  x°. 

b.)  This  is  also  the  form,  to  which  the  principles  of  §  200 
apply,  and  will,  therefore,  be  commonly  employed  in  our 
future  discussion  of  the  subject. 

§  205.   In  solving  the  complete  equation, 

x2-\-2px-\-q2  =  0, 
we  may  happen  to  have  q  =p.     In  this  case,  the  equation 
becomes 

x2-\-2px-]-p2  =  0, 

or  (§  93.  L),  (x+P)  (x+p)  =  0. 

We  have  here  the  equation  resolved  into  two  binomial 
factors  (§  200.  1),  either  of  which  may  be  equal  to  zero. 
But  in  this  case,  the  factors  are  equal ;  and,  consequently, 
the  values  of  x,  found  from  them,  will  be  equal.  The 
equation  is  said,  in  this  case,  to  have  equal  roots,  viz.  —p 
and  —p. 

Thus,  let     »2+20ar+100  =  0. 
Then  (x-f-10)(ar+10)  =  0,  and  x  —  —10,  or  —10. 

If  we  had  x2— 20x-4-100  =  0, 
we  should  have  x  =  +10,  or  -[-10. 

a.)  The  sum  of  the  roots,  with  their  signs  changed  is  still 
equal  to  the  coefficient  of  x1,  and  their  product,  to  the  co- 
efficient of  x°. 


§  206,  207.]  COMPLETE  EQUATIONS.  157 

§  206.  But  suppose  that  q  is  not  equal  to  p ;  i.  e.  that 
q2,  the  coefficient  of  a;0,  is  not  equal  to  p2,  the  square  of 
half  the  coefficient  of  a:1. 

Then  the  equation,  x2-\-2px-\-q2  =  0, 
gives  x  ~-\-2px  =  — q  2 . 

If  now  we  add  p2  to  hoth  sides  of  the  equation,  the  first 
member  will,  evidently,  become  a  trinomial  perfect  square 
(§§  89,  172),  and  we  shall  have 

x2-{-2px-\-p2  =p2 — q2. 

x-\-p  =  J(p2—q2);  §52.  N. 

and  x  =  —p-\-*/(2*2 — 92)'  or  x  =  ^p — */{p2 — ?2)- 

Thus,  let  x2+8:c+15  =  0. 

Then  cc2+8;c  =  — 15. 

Adding  42(=j»2),       x*-+8x+l6  =  — 15+16  =  1, 
Extracting  the  root,  a+4  =  ±1. 

x  =  — 4±1  = — 3,  or — 5. 
x+3  =  0,  or  x-\-o  =  0  (§  199.  b) ; 
and    (a+3)0+5)  =  a:2+8.z+15  =  0  (§§  200,  208.  b). 

Also       (—3)  2+8(— 3)+15  =  9—24+15  =  0 ; 
and  (— 5)  2+8(—5)+l  5  =  25—40+15  =  0. 

The  process  of  rendering  the  first  member  a  perfect 
square,  is  commonly  called  completing  the  square. 

Hence  we  have,  for  solving  a  complete  equation  of  the 
second  degree,  containing  but  one  unknown  quantity,  the 
following 

RULE. 

§  207.  1.  Reduce  the  equal  ion  to  the  form  x2±2px 
±q-  —  0.  Transpose  the  coefficient  of  x°  to  the  sec- 
ond member,  and  add  the  square  of  half  the  coefficient 
ofx1  to  both  sides. 

2.  Extract  the  square  root  of  both  members,  ana 
solve  the  equation  of  the  first  degree  thus  obtained. 

ALG.  14 


158  EQUATIONS  OF  THE  SECOND  DEGREE.         [§  20B'.- 

1.  Given  x-+ix—  60  =  0,  to  find  x. 

2.  Given  x-— Oa+lO  =  65,  to  find  x. 

3.  Given  3a:2— 3a:+9  =  8£,  to  find  x. 

_4hs.  a;  =  §,  or  J* 

4.  ^2_ia;+30^  =  52i,  to  find  a;. 

Ans.  x  =  7,  or  — 6 -*-. 

§  208.  a.)  The  same  effect,  obviously,  will  be  produced, 
if,  without  transposing  the  coefficient  of  x°,  or  the  absolute 
term  as  it  is  sometimes  called,  we  add  to  both  sides  a  quan- 
tity, tvhich  together  with  that  coefficient  shall  be  equal  to  the 
square  of  half  the  coefficient  ofx1,  (i.  e.  p2 — q2).     Thus, 

x2+2px+q2+(p2-q2)  =  p2-q*  ; 
or  x2-\-2px-\-p2  =p2 — q2. 

x+p  =  ±^/(p2—q2).  §52.  N. 

x  =  — p++/(2^— ?2)>  or  x  =  — p— «/(p2—  q*). 
b.)  These  values  (§§  206,  208)  give  the  equations 

x+p—(p2—q2)2=-Q,  and  x+2>Mp-—  ?2)2=- 0. 

[x+p-ljjS-q^lx+ji+iP*-?2)^  = 

(a+p)2—  [{p-— ?2)2]2  (§  92)  =  xJ+2px+p2—p2+q2  = 
x2-\-2px+q2  =  0.  §  200.  1 . 

Also  p-(p2-q"f+p+lP°—V2f=2p->   §200.3. 

and      [- p—{pn—  <72)2][— p+(j*2— 72)"]  —  <?2-     §200.4. 

1.  Given  a:2-f-6a+8  =  0,  to  find  x. 

Here  ?2  =  8,  2p  =  6  ;  .-. p  —  3,p2  =  9,  andjo2— q2  =  i . 
Hence  we  have     x2-\-Gx-\-9  =  1 ;  and  x-\-3  =  ±1. 
x  =  —  3+1  =  —  2,  or  x  =  —  3— 1  =  —  4. 
jc— (— 2)  =  x-\-2  =  0,  or  x— (—4)  =  a+4  =  0. 
Hence  (a+2)(a>H)  =  a;2+6aH-8  =  0.  §200.1. 

Also         2+4  =  6  =  2;?,  and  2X4  =  8  =  q*.     §200.3,4. 

2.  Given  x2— <6x— 40  =  0,  to  find  a\ 

Ans.  x  =  10,  or  — 4. 
Here  ^2— $2  =  9_(_40)  =49. 


§  209.]  COMPLETE  EQUATIONS.  159 

3.  Given  x- — 16a;-f-63  =  0,  to  find  x. 

4.  Given  ar2+16x+63  =  0,  to  find  x. 

§  209.  c.)    Resume  the  equation  Ax2-\-Bx-\-C=0  ; 
or  Ax2-\-Bx  =  —  C. 

-Multiplying  by  A,     A2x2-\-ABx  =  —A  C. 
Adding  (%B) 2,    A2x2+ABx+\B2  =  \B2— A C. 

Extracting  the  root,     Ax+\B  =  (IB2— A  0)K 

Hence,  to  complete  the  square, 

Reduce  the  equation  to  the  form  Ax2-\-Bx-{-C=  0 ;  trans- 
pose the  coefficient  of  x°  ;  multiply  by  the  coefficient  of  x2  ; 
■and  then  add  to  both  sides  the  square  of  half  the  primitive 
coefficient  ofx1. 

1.  Given  5x2+4x— 204  =  0,  to  find  x. 

5x2-\-lx=2Qi. 
25x2+20o:4-4=:1024. 
ox+2  =  ±32  ;  and  .-.  5x  =±  30,  or  —34. 
x  =  6,  or  — 6f . 

2.  Given  2x2-\-8x— 90  =  0,  to  find  x. 

Ans.  x  =  o,  or  — 9. 

d.)   Or,  Ax2+Bx+C=z0. 

Multiplying  by  A,    A2x2+ABx+A C—  0. 
Adding  ±B2—AC,    A2x2+ABx+\B2  —  \B2—AC. 

Hence,  to  complete  the  square, 

Multiply  the  equation,  Ax 2-\-Bx-\-  O=0,  by  A,  and  add 
■to  both  sides  \B2—AQ. 

Given  3a:2-f-2x— 85  =  0,  to  find  x. 

9x2+6z— 255  =  0. 

Here       \B2—A  C=  1— (-255)  =  256. 

9x2+6x+l  =  256. 

.-.     3as-|-l  =  ±16;  and.-.  3x  =  — 1±16  =  15,  or— 17. 


2 


x  ■=.  5,  or  — 5^ 

Note.    When  ^  =  l,  this  solution  (§209.  c,  d)  is,  evidently, 
the  same  as  that  of  §§  207,  208. 


160  EQUATIONS  OP  THE  SECOND  DEGREE.         [§  210. 

§210.  e.)  Again, 

Ax*+Bx-\-C=0;  or  Ax2-\-Bx  =  —  01 
Multiplying  by  A  A, 

4A*x2+4ABx  =  —4A  C. 
Adding  B*,     4A2x2-\-4ABx-\-B2  =  B^—iAC. 

2Ax-{-B  —  (B  »—4A  C  )  *. 

_  —  B±^(B2—4AC) 

.*.  x  — — — . 

2A 

Hence,  to  complete  the  square, 

Reduce  the  equation  to  the  form  Ax2-\-Bx-\-Cz=i  0  ;  trans- 
pose the  coefficient  of  x°  ;  multiply  by  four  times  the  coeffic- 
ient of  x2  ;  and  add  to  both  sides  the  square  of  the  primitive 
coefficient  of  x1. 

1.  Given  3a;2— 3a+§  =  0,  to  find  x. 

36a;2— 36a;  =  —  8. 
36*2—  36a+9  =  —8+9  ==  1. 
6a;— 3  =  ±1 ;  .-.  Gx  =  3±1  =  4,  or  2. 
x  =  | ,  or  £. 

2.  Given  \x2—: \x— 22£  ==  0,  to  find  x. 

Ans.  x  =  7,  or  —  6J» 

/.)  Or,  multiply  the  equation,  Ax2-\-Bx-\- (7=0,  by  4A 
Then  4A2x2+4ABx-{-4AC=Q. 

Adding  B2—4AG, 

4A2x2-\-4ABx-{-B2  =  B*—4A  C ;  as  in  e  above. 
Hence,  to  complete  the  square, 

Multiply  the  equation,  Ax2 +#£+(7=0,  by  4A  ;  and  add 
to  both  sides  B'2—4AC. 

Given  x2 — 5x — 24  =  0,  to  find  x. 

4x2—  20a;— 96  =  0. 
Here     B2— 4A  C  =  25— (—96)  =  25+96  =  121, 
4a;2— 20a+25  =  121. 
2a:— 5  =  ±11;   .-.a;  =  8,  or — 3. 
Note.    When  A  —  1,  this  solution  (§210.  e,f)  is,  obviously, 
the  same  as  that  of  §§  207,  208. 


§211-213.]  GENERAL  DISCUSSION.  161 

§  211.  Let  x2-\-2px-{-q2  =  0  be  any  equation  whatever 
of  the  second  degree,  containing  but  one  unknown  quanti- 
ty ;  let  also  a  be  one  of  its  roots  (i.  e.  such  a  quantity  as, 
being  substituted  for  x  in  the  given  equation,  will  make  the 
members  equal ;  or,  in  other  words,  will  reduce  the  first 
member  to  zero).     See  §  39. 

1.  Since  a  is  a  root  of  the  equation,  we  have  x  =  a,  and 
x — a  —  0. 

Divide  the  given  equation  by  x — a. 
Thus  x2-\-2px-\-q< 


x*2 — ax 


x — a 


x-\-{a-\-2p) 


(a-\-2j))x 

(a-{-2p)x — a  2 — 2pa 

a2-\-2pa-\-q2  =  0,       because 
the  remainder  is  simply  the  first  member  of  the  given  equa- 
tion with  a  substituted  for  x ;    which,  by  hypothesis,  redu- 
ces it  to  zero.     The  division  is  therefore  perfect  (§  82.  g). 
Hence     (x—a)  (x+a+2p)  =  x  2-\-2px-\-  q  2  =  0.     §200.1. 

2.  And  the  equation  will  be  satisfied,  if  we  take  x — a 
=  0,  or  x-\-2p-\-a  =:  0  (i.  e.  if  x  —  a,  or  x  =  — 2p — a  =  b 
(by  substitution).     §  200.  2. 

3.  We  have  also     —  a-\-(2p-\-a)  =z  2p.  §  200.  3. 

4.  Moreover,  since  a2-\-2pa-\-q2  =  0,  (see  1,  above), 

a(— a— 2p)[  =  —a2—2pa']  =  q2.         §  200.  -J. 

§  212.  5.  It  is  also  evident  from  §  211.  1,  that,  if  a  is  a 
root  of  the  equation  x2-\-2px-\-q2  =.  0,  this  equation  is  di- 
visible by  x — a,  and  will  give  a  quotient  of  the  form  x — l>, 
of  which  the  second  term  is  the  other  root  with  its  sign 
changed. 

§  213.   Hence,  universally  (§  200), 

1.  Every  equation  of  the  second  degree,  of  the  form 
x2±2])x±q2  =  0,  containing  but  one  unknown  quantity, 
can  be  resolved  into  two  binomial  factors,  of  the  first  degK  e 

*14 


162  EQUATIONS  OF  THE  SECOND  DEGREE.  [§  214-216. 

in  respect  to  x  (§  28.  b) ;   either  of  which,  being  put  equal 
to  zero,  gives  a  root  of  the  equation. 

2.  Every  such  equation  has,  of  course,  two  roots. 

3.  The  algebraic  sum  of  the  roots,  with  their  signs 
changed,  is  always  equal  to  the  coefficient  of  a;1. 

4.  The  product  of  the  roots  is  always  equal  to  the  coef- 
ficient ofx°. 

5.  Every  such  equation,  of  which  a  \s  a  root,  is  divisible 
by  x — a. 

§  214.  a.)  Hence  (§  213.  3,  4), 

Cor.  I.  (1.).  If  the  coefficient  of  x1  be  equal  to  zero,  the 
roots  must  be  numerically  the  same,  but  with  opposite  signs 
(§  199.  a).  (2.)  If  the  coefficient  of  x°  be  equal  to  zero,  one 
of  the  roots  must  be  zero  (§  203.  a). 

§215.  b.)  Also  (§§213.  4;  9.  a;  213.3), 

Cor.  ii.  (1.)  If  the  coefficient  of  x°  be  positive,  the 
roots  must  have  like  signs ;  (2.)  if  negative,  unlike. 
(3.)  If  the  two  roots  have  the  same  sign,  it  will  be  unlike  the 
sign  of  the  coefficient  of  x1.  (4)7/'  they  have  different 
signs,  the  sign  of  the  root  which  is  numerically  the  greater 
will  be  unlike  that  of  the  coefficient  of  x1. 

c.)  It  is  obvious,  that,  if  the  roots  have  like  signs,  the  co- 
efficient of  x1  will  be  numerically  equal  to  their  arithmeti- 
cal sum  ;  and,  if  they  have  unlike  signs,  to  their  arithmeti- 
cal difference. 

§  21 6.  d.)  If  q-  be  positive  and  greater  than  p2,  the  pro- 
duct of  two  numbers  is  required  to  be  greater  than  the 
square  of  half  their  sum.  This  will  be  shown  to  be  impos- 
sible ({  220.  b),  and  as  no  real  numbers  can  satisfy  thia 
condition,  the  roots  will  be  imaginary  (§§  201  ;  217.  I.). 
Hence, 

Cor.  in.  If  the  coefficient  of  x°  be  positive  and  greater 
than  the  square  of  half  the  coefficient  of  x1,  the  roots  must 
be  imaginary. 


§  217,  218.]  SIGNS  OP  THE  ROOTS.  163 

§  217.  e.)  The  above  principles  may  be  otherwise  de- 
monstrated ;  thus, 

I.  Let  q2  he,  positive. 

Then  x2±2px-{-q2  =  0  ;  and  x  —  ^p±^{p2^q2). 

Now,  evidently,  *J(p2 — ?2)<p;  and,  therefore,  both  the 
roots  are  negative,  when  2p  is  positive  ;  and  positive,  when 
2p  is  negative  (§  215.  1,  3). 

It  is  also  evident,  that,  if  q2>p2,  «/(p2 — q2  is  imagina- 
ry(§§28./.2;  216.). 

It  is  also  manifest,  that,  if  one  of  the  roots  is  imaginary, 
both  must  be. 

II.  Again,  let  q2  be  negative  Then  x2±2px — q2  =  0  ; 
and  x  =  ^p±«/(p2-\-q2). 

Here,  obviously,  .y(.P24~!72)>/>  5  and,  therefore,  one 
root  must  be  of  the  same  sign  as  p ;  and  the  other,  different 
(§215.  2). 

Also,  the  root  which  is  of  the  same  sign  as  p  (i.  e.  of  a 
sign  different  from  2/>  on  the  other  side},  will,  of  course,  be 
numerically  the  greater  (§  215.  4). 

§  218.  f.)  Determine  whether  the  signs  of  the  roots  in 
the  following  equations  are  like  or  unlike  ;  if  like,  whether 
positive  or  negative ;  and,  if  unlike,  which  is  numerically 
the  greater.  Also  determine  whether  any  of  these  equa- 
tions have  imaginary  roots. 

1.  x2+21x+110  =  0;  x2— 20+75  =  0. 

2.  x2— 23x4-130  =  0  ;  x24-23x-fl30  =  0. 

3.  x2±60x-|-1000  =  0;  x 2±60x— 1000  =  0. 

4.  x2±G0x— 11200  =  0;  x2±10x  =  200. 

g.)  1.   Write  the  equation,  of  which  3  and  4  are  the  roots. 
Ans.  (x— 3) 0—4)  =  x2— 7x4-12  =  0. 

2.  Write  the  equation,  whose  roots  are  — 3  and  — 4; 
—11  and 4-20;  -4-11  and— 20;  —10  and  -4-10  ;  —10  and 
—  10;  10  4V— 5  and  10— y— 5;  — 64~5y— 1  and 
—6— 5y— 1. 


1  C4         EQUATIONS  OF  THE  SECOND  DEGREE.  [§  219,  220. 

h.)  In  the  last  example,  we  have  (§§  92,  162) 
(T+6_5y_l)(.r+6+5y-l)  =  (r+6)2-(5y-l)9  = 

^._[_6)2_j_52  =  0;   which  is,  evidently,  impossible  (§201. 

N.  2). 

§  219.  ».)  Again  (§  210),  we  shall  have  the  value, 

e  =  — ^Li — -,  real,  when  B- — 4  J.  C  is  positive  ; 

2A 

and  imaginary,  when  i?2— 4J.  C  is  negative.     That  is,  the 

roots  will  be  real  and  unequal,  when  B- — AA  C>0  ; 

real  and  equal,  "      B°—  \A  C—0; 

imaginary,  "      -B2 — 4-4  (7<0. 

PROBLEMS. 

§  220.  1.  Given  a;2— 2x— 24  =  0,  to  find  the  values  of 
x#  Ans.  x  =  -4-6,  and  — 4. 

2.  Given  a:2-f-l  2x4-35  =  0,  to  find  x. 

Ans.  x  =  — 5,  Or  — 7. 

3.  Given  3x 2+2x— 10=75,  to  find  a-. 

_4ws.  jc  =  5,  or  — of. 

4.  Given  x°—  x— 210  =  0,  to  find  a. 

_4rcs.  a;  =  15,  or  — 14. 

5.  Given  £x2— £x+6f  =  7,  to  find  a:. 

_4ws.  a;  =  1-i-,  or  — {?. 

6.  Find  two  numbers  whose  sum  is  100,  and  whose  pro- 
duct is  2100. 

Let  x  =  one  of  the  numbers. 
Then  100 — x  =  the  other  ; 

and  x(100 — x)  =±  2100,  by  the  second  condition. 

a;2— 100x  =  —  2100. 

We  might  have  formed  this  equation  immediately  by 
considering,  that  the  sum  of  the  required  numbers  taken 
with  a  contrary  sign  must  be  equal  to  the  coefficient  of  x ' ; 
and  their  product,  to  the  coefficient  of  a:0. 

Thus  x*— 100x+2100  =  0. 


§  220.]  PROBLEMS.  165 

x2— 100;r-{-2500=:400.  §208* 

x  =  70,  or  30. 
Otherwise,  let  x  =  the  excess  of  the  greater  number 
above  50  (i.  e.  half  the  sum  of  the  numbers) ;  then  50-}-x  = 
the  greater,  and  50 — x  =  the  less. 

Hence      [(50-f-.r)(50— x)  — ]2500— x2  =  2100. 

x2  =  400;  and  a:  =  ±20. 
50-f-a:  =  70,  or  30  ;  and  50— x  =  30,  or  70. 

7.  Find  two  numbers,  whose  sum  is  100,  and  whose 
product  is  24Q0. 

8.  Find  two  numbers,  whose  sum  is   100,  and  whose 
product  is  2500  (§  205). 

9.  Find  two  numbers,  whose  sum  is   100,  and  whose 
product  is  2600  (§216). 

Am.  50+10y— 1,  and  50—10^—1. 

10.  Find  two  numbers,  whose  sum  is  S,  and  product  P. 


Am. 


fK?~P)*  and  4-  (f-p)* 


a.)  In  what  case  will  these  values  be  imaginary  ? 

Am.  When  P  >  ^T  =  (-^)  '1.     See  9,  above. 

Hence, 

The  product  of  two  numbers  can  never  be  greater  than 
the  square  of  half  their  sum. 

b.)  This  principle  can  be  proved  otherwise ;  thus, 
Let  S  be  the  sum  of  two  numbers,  and  D,  their  differ- 
ence. 

Then  l<S+l&  =  the  greater,  §  57.  3. 

and  IS — \D=.  the  less.  §  60.  4. 

Also  (|£4-i #)(££—  \D)  =  (±S)2—(W)2  =  their  pro- 
duct ;  which  is  obviously  greatest  when  (\D)  2  is  least,  i.  e. 
when  %D=Q. 

11.  The  algebraic  sum  of  two  numbers  is  8,  and  their 
product  is  — 240.     What  are  the  numbers  ? 


166  EQUATIONS  OF  THE  SECOND  DEGREE.  [§220. 

Here,  by  §  200,  3,  4, 

x2— 82—240  =  0,  or  x2— 8x=  240. 
a:  =  20,  or— 12. 
Verification.  If  one  of  the  numbers  is  20,  the  other  is 
8— 20  =—12  ;  and  20X— 12  =— 240.  Or,  if  one  of  the 
numbers  is  —12,  the  other  is  8— (—12)  =  20,  &c.  Or,  if 
one  of  the  numbers  is  20,  the  other  is  —240-^20=— 12, 
and— 12+20  =  8. 

12.  Find  two  numbers,  whose  difference  is  10  ;  and  such 
that,  if  600  be  divided  by  each  of  them,  the  difference  of 
their  quotients  shall  be  10. 

13.  Find  a  number,  which  added  to  its  square  makes 
12.  Ans.  6,  or  —7. 

14.  Find  two  numbers,  whose  sum  is  16,  and  the  sum 
of  whose  squares  is  130.  Ans.  7  and  9. 

15.  "What  two  numbers  are  there,  whose  sum  is  S,  and 
the  sum  of  whose  squares  is  Q  ? 

Ans.  ^S+U(2Q—S*),  and  \S—  W(2Q— &)• 

c.)  "When  will  these  results  be  imaginary  ? 

Ans.  "When  S2>2  Q.     Whence, 

The  square  of  the  sum  of  two  numbers  cannot  be  greater 

than  twice  the  sum  of  their  squares. 

Note.  As  either  of  the  numbers  may  be  negative,  this  applies 
equally  to  the  square  of  the  difference. 

16.  The  sum  of  two  numbers  is  25,  and  the  sum  of  their 
cubes  is  8125.    "What  are  the  numbers?    Ans.  20  and  5. 

17.  A  rectangular  field  contains  20  acres,  and  one  side 
is  40  rods  longer  than  the  other.  "What  are  the  dimensions 
of  the  field  ?  Ans.  80  rods  long,  and  40  wide. 

18.  A  rectangular  park,  60  rods  long  and  40  wide,  is 
surrounded  by  a  street  of  uniform  width,  containing  1344 
square  rods.     How  wide  is  the  street? 

Ans.  6  rods,  or  — 56  rods. 
d.)  The  second  value,  — 56,  is  clearly  not  a  proper  solu- 
tion to  the  problem ;   but  it  is  a  root  of  the  equation,  and. 


§221.]  EQUATIONS  OF  THE   2)lth  DEGREE.  1G7 

in  some  sense,  satisfies  the  conditions  of  the  problem.  For 
we  find  the  area  of  the  street  by  multiplying  its  width  by 
each  of  the  sides  of  the  park,  and  adding  to  the  sum  of 
these  product  5  the  squares  formed  at  the  four  corners. 
Thus, 

6X60      ;xG0+Gx40-}-6x4(4-4xC2  =  1344. 
So        2(-  +-?(—56x40)+4(— 56)2  =  1344. 

It  freqi  lappens,  as  we  have  already  seen  (§137  . 

that  the  a>_  expression  of  a  problem  is  more  general, 

and  admits  of  more  solutions,  than  the  problem  itself  as 
expressed  in  ordinary  language. 

x*n+Pxn-\-Q=0. 
§  221.   The  preceding  methods  apply  not  only  to  equa- 
tions of  the  second  degree,  but  to  all  equations  of  the  form 

x*-"+Pxn-\-Q  =  0, 
in  which  the  unknown  quantity  appears  in  only  two  term? : 
and  its  exponent  in  one  of  the  terms  is  double  that  in  the 
other. 

This  equation  may  be  put  under  the  form, 

(xn)  2+/V-f  Q  =  0. 
Completing  the  square  (§  207), 

O")  2-f  P^'-fiP2  =  i?2_  Q. 

xn  =  —^P±(\P^—Q)^. 

x  =  t—hP± (IP--  Q) 2]".  §  52.  N. 

1.  Given  x*— 52x2+576  =  0,  to  find  x. 

Jlns.  x=  ±6,  or  ±4. 

2.  Given  Xx — \*/x  =-\\,  to  find  x. 

Ans.  */x  =  o,  or  — H;  .-.  x  =  9,  or  2|. 
In  verifying  these  results,  Jx  must  be  positive  for  the  first  valua, 
and  negative,  for  the  second.     A  similar  remark  applies  to  the  fol- 
lowing example. 

3.  Given  (ic-f-12)*+(x-f-12)^=  6,  to  find  x. 

(*+12)-2-+(*-hl2)H-i  - 1+6  =  ~. 


il 


168  EQUATIONS  OF  THE  SECOND  DEGREE.  [§  222. 


1 


OH-12)4  =—  \±%—%  or  —  3. 

x+12  =  lG,  or  81.  §52. 

x  =  4,  or  69. 

4.  Given  x^+xS  =  756,  to  find  x. 

Ans.  x  —  243,  or  (—28)*. 

3 

5.  Given  x3 — x2  =  56,  to  find  x. 

^4ws.  x  =  4,  or  ( — 7)^. 

Note.  We  have  seen  (§  213.  2)  that  every  equation  of  the  sec- 
ond degree  has  two  roots.  It  will  be  proved  hereafter,  that  every 
equation  has  as  many  roots  as  there  are  units  in  its  degree.  Sec 
1,  above.  The  above  process,  however,  does  not  always  exhibit  all 
the  roots. 

§  222.  If  an  equation  contain  radicals  which  cannot  be 
treated  by  the  method  of  §  221,  it  may  frequently  be  re- 
duced by  properly  arranging  the  radical  terms  containing 
the  unknown  quantity,  and  raising  both  members  to  the 
requisite  power.  There  is  frequently  great  advantage  al- 
so in  rendering  a  binomial  surd  rational  (§§  186,  187). 

The  radicals,  which  most  frequently  occur,  are  radicals  of  the 
second  degree  (i.  e.  expressions  of  the  square  root  of  quantities). 

1.    Given  x-\-J (2ax-\-x2)  =  a,  to  find  x. 

We  have         */(2ax-\-x-)  =  a — x. 

Then  squaring  2ax-\-x2  =  a2 — 2ax-\-x2. 

Aax  z=z  a2  ;  and  x  =l  \a. 

Clearing  of  fractions,  x-\-a-\-2ls/(ax)  z=  b2x. 

Extracting  the  square  root, 

•v/r-j-^/a  —  ±  b^/  x ; 

or  (lzfb)^/xz= — „/«. 

(lj:b)2x  =  a. 
a  a 

-(i^by-'^iy' ' 


§223.]    RADICALS— TWO  UNKNOWN  QUANTITIES.  169 

3.  Given  2x+2y(a*+x*)  =  _-JL_,  to  find  x. 

Ans.  x  —  \a. 

_.         Jx-\-J(x— a)        n-a 

4.  Given  — — ]-^-j {■  = ,  to  find  x, 

Jx — +/{x — a)      x — a 

If  we  render  the  denominator  of  the  first  member  ration- 
al (§  187),  multiply  by  a,  and  extract  the  square  root,  we 

shall  have 

±na 

Clearing  of  fractions  and  transposing, 

*/(x2 — ax)  =  a±na — x  =  (l±n)a — x.. 
Squaring,        x*— ax=z  (l±n)2a*—2(l±n)ax+x*. 

{l±ny-a 
l±2n  ' 

k     ^.         +/ (a-\-x)-\-+/ (a—x)  , 

5.  Given     ,;         (       ,; (=J,  to  find  *. 

V  («+#)  — y  (a— a:) 

2a5 

6.  Given  ^+il±^M  =  9,  to  find,. 

7.  Glven_______i__  =  __,tofind 

the  value  of  a:.  Ans.  x  —  ±\. 

§  223.  Every  complete  equation  of  the  second  degree, 
containing  two  unknown  quantities,  and  having  only  posi- 
tive integral  powers  (§  22.  c,  d),  is,  obviously,  of  the  form 
(§197) 

A^+Bxp-\-Cx3+J)i/+JEx-{-F=  0. 

That  is,  it  contains  terms  of  the  zero,  the  first,  and  the 
second  degree  with  respect  to  both  and  each  of  the  un- 
known quantities. 

a.)  A  single  equation  of  this  kind  is,  of  course,  indeter- 
minate (§  122.  a) ;  and  will  give,  for  any  value  whatever  of 
alg.  15 


170  EQUATIONS  OF  THE  SECOND  DEGREE.         [§  224. 

itlier  of  the  unknown  quantities,  two  values  of  tlie  other 
13.  2). 

§  22  4.  h.)  Such  equations  are  of  continual  use  in  tha 
;er  applications  of  Algebra,  in  expressing  the  relation 
rreen  two  variable  (§  136)  quantities  which  are  so  con- 
nected, that  a  change  in  the  value  of  one,  in  general,  in- 
volves a  change  in  the  value  of  the  other;  i.  e.  between 
two  variables,  which  are  functions,  one  of  the  other  (§§  2G. 
136.  a). 

c.)  Thus,  let  x  denote  the  distance  from  any  point  in  the 
circumference  of  a  circle  to  a  given  straight  line,  and  y  the 
distance  from  the  same  point  to  another  line  perpendicu- 
lar to  the  first.  Then  the  relation  between  these  distances 
will  be  such,  that,  if  one  of  them  be  given,  the  other  will 
be  determined ;  and  if  another  point  be  taken  at  a  different 
distance  from  the  first  line,  it  will  also,  in  general,  be  at  a 
different  distance  from  the  second.  That  is,  a  particular 
value  of  x  requires  a  corresponding  value  of  y ;  and  a 
change  in  the  value  of  x  involves,  in  general,  a  correspond- 
ing change  in  the  value  of  y. 

d.)  An  equation,  expressing  some  known  relation  be- 
tween these  distances,  is  called  an  equation  of  the  curve. 
By  means  of  such  an  equation,  the  properties  of  the  curve 
are  easily  and  rapidly  deduced. 

e.)  The  equations  of  the  circle,  ellipse,  parabola  and  hy- 
perbola are  of  the  second  degree,  and  contain  two  variables. 

Thus,  y2-\-x2 — R-  =  0  is  the  equation  of  the  circumfer- 
ence of  a  circle,  when  the  distances  x  and  y  are  measured 
from  two  diameters  at  right  angles  to  each  other.  For  in 
that  case  these  distances  for  any  point  of  the  curve,  togeth- 
er with  the  radius  drawn  to  that  point,  form  a  right  angled 
triangle,  of  which  the  radius  is  the  hypotenuse.  Whence 
x*-+y2  =  B*  (Geom.  §188). 

Note.  A  straight  line  is  represented  by  an  equation  of  the  frst 
degree,  between  two  variables.  Tliua,  y  =  ax+b;  a  and  b  being 
either  positive  or  negative. 


j  2*25,  226.]     two  unknown  quantities.  171 

f.)  The  employment  of  equations  of  this  kind  for  the  d 
co very  of  geometrical  truth  helongs  to  Analytical  Geome- 
try and  to  the  Differential  and  Integral  Calculus.     1 
this,  at  the  same  time,  furnishes  one  of  the  most  important 
applications  of  the  principles,  already  demonstrated,  of  equa- 
tions of  the  second  degree. 

§  225.  The  ordinary  algebraic  treatment  of  equations  of 
the  second  degree,  containing  two  unknown  quantities,  sup- 
poses two  equations  (§  122.  d,  e) ;   and  deduces  values 
the  unknown  quantities,  which  will  satisfy  hoth  equatio 

Let  there  be  given  the  two  equations, 

Ay*-\-Byx+Cx*+Dy-\-Ex+F=  0, 
and  ATjr»+Bjtx-t-  C'x2-\-D'y-\-F'x-\-F'  =  0. 

If  now  one  of  the  unknown  quantities,  as  y,  be  found  in 
terms  of  x  and  known  quantities,  and  this  value  be  sub 
tuted  in  the  other  equation,  there  will,  of  course,  result  an 
equation  containing  but  one  unknown  quantity.  If  this 
equation  be  solved,  and  the  values  found  for  x  be  substitu- 
ted in  one  of  the  primitive  equations,  corresponding  values 
of  y  may  be  found. 

But  it  is  sufficiently  evident,  that  the  equation  so  obtain- 
ed by  the  elimination  of  one  of  the  unknown  quantities  will 
be  of  the  fourth  degree,  which,  in  its  general  form,  we 
not  yet  prepared  to  solve. 

§  226.   Though  we  are  not  prepared  for  a  general  sol 
tion  of  two  equations  of  the  second  degree  containing  two 
unknown  quantities,  yet  certain  classes  of  such  equation.-!, 
can  be  solved  by  applying  the  principles  already  demon- 
strated. 

This  is  true  of  all  those  equations,  in  which  the  elimina- 
tion of  one  of  the  unknown  quantities  results  in  an  equa- 
tion either  of  the  second  degree,  or  of  the  form,  xin±Px 
±0  =  0  (§221). 

1.    Given  x2+r  =  100, 

x°~—7ji  =  28,  to  find  x  and  y. 


172  EQUATIONS  OF  THE  SECOND  DEGREE.        [§  226. 

Adding,  subtracting,  and  dividing  by  2,  we  bare 

•  a:2  =  64,  .\*=±8;. 
and  ^2  =  36,  .-.  y=±&. 

2.   Given  z2-fy2  =  100, 

xy  =  48,  to  find  x  and  y. 
From  tbe  second  equation, 


/48\s 
Substituting  in  tbe  first, 


••-©' 


^+^=10* 


a:*— 100a:2:=  —  2304.  §221. 

a2  =  64,  or  36 ;  and  x  =  ±8,  or  ±6. 
y  =  ±6,  or  ±8. 
a.)  Tbe  last  example  may  be  more  conveniently  solved 
witbout  elimination.     Tbus,  adding  and  subtracting  twice 
tbe  seeond  equation  to  and  from  tbe  first,  we  bave 

attd  *»-2ay+y9  =  4J 

x-\-y  =  ±  1 4 ;  and  ic— y  =  ±  2. 
a:  =  ±8,  or  ±6 ;  and  y  =  ±6,  or  ±8  ;  as  before. 

3.  Given  x2-\-xy-\-y2  =  112, 

ar2_a:y_|_y2  —  43^  to  find  x  and  y. 

Ans.  x  =  ±8,  y  =  ±4. 

4.  Given  a:2+a#  =  180, 

xy-\-y2  =  45,  to  find  ar  and  y. 

^4ns.  a:  =  ±12,  y=:±3. 

5.  Given  4ry  =  96— x2y2, 

x-\-y  —  6,  to  find  x  and  y. 

Ans.  ar  =  4,  or  2,  or  3±y21 
y=2,  or  4,  or  3:fV2L 

6.  Given  ar2+a;4-y  =  18—y2' 

a:y  =  6,  to  find  a?  and  $, 

Ans.  x  =  3,  or  2,  or  —  3±<y3, 
y  =  2,  or  3,  or  —  3:f  </3. 


§  227,  223.]     two  unknown  quantities.  173 

§  227.  b.)  If  one  of  the  equations  be  of  the  first  degree, 
and  one  of  the  quantities  be  eliminated,  then  the  resulting 
equation  will  be  only  of  the  second  degree. 

1.  Given  2x+y=z  10, 

2x2 — xy-\-'3y-  =  54,  to  find  x  and  y. 
From  the  first,     y  =  10— 2x. 
Substituting,    2a;2— a(10— 2ar)+3(10— 2x)  *  =  54. 

Ans.  x  =  3,  or  5  J- ;  y  =  4,  or  — f . 

2.  Given  2x-\-y  =  9, 

;zy  =  10,  to  find  x  and  y. 

Ans.  x  =  2,  or  2?> ;  #  =  5,  or  4. 

3.  Given  x-\-y  =  10, 

a;2_j_^2  =  505  to  find  a;  and  y. 

$  228.  c.)  It  is  sometimes  convenient  to  employ  auxilia- 
ry unknown  quantities,  such  as  the  sum  and  difference,  or 
the  sum  or  difference  and  product  or  quotient. 

Note.  If  one  or  both  of  the  equations  be  of  a  higher  degree,  the 
problem  can  frequently  be  solved  by  an  equation  of  the  second  de- 
gree. 

1.    Given  x-\-y  =  a, 

x3-\-y3  =  b,  to  find  x  and  y. 

Let  x  =  s-{-t,  y  =  s— t,  and  .*.  (§  57.  3)  s  =  ^(x-\-y)  =  \a. 

Then  x3+y3  =  (s+t)  3-f(s— t)  3  =  2s3+0si2  =  b. 

„  b—2s3        ,  ,  /&— 2s3\i 

Hence  £2  =  — —  ,  and  t  =  ±  ( )  . 

6s      '  V     lis    -/ 

...       *  =  S±(_-)    ,  and^  =  Sq:(— -)   . 

.•.,  introducing  the  value  of  s, 

,  _  «.  /^-^«3  \i  _  «.  /46-«3x  i 

■C_2±v_3^  )  -^vi^r) ' 

_  «     /Ab — a3\l 

and        ^2ni^-J- 

Let  a  =  10,and  6  =  370;  a  =  12,and  6=1008;  a  =  7} 

and  b  =  217. 

*15 


174  EQUATIONS  OP  THE  SECOND  DEGREE.        [§228. 

2.    Given  x-\-y  =  8, 

x*-\-y*  =  70 6,  to  find  x  and  y. 
Let  x  =  s-^-t,  y  =  s — t,  and  .*.  s  =  l(x-\-y)  =  4. 
Then  z*+y*  =  (5+*)4-f(fi_*)4  —  2s*-f-12s2i!2+2^  =  706. 
Or,  as  s  =  4,        512+192*2+2^  =  706. 

*4+96*2r=97. 
<2  =  1,  or  — 97 ;  and  t  =  ±l,  or±y— 97. 
x  =z  5,  and  y  =  3;  or  a?  =  3,  y  =  5  ; 
or  xz=  4±y— 97,  and  y  =  4qV— 97. 

3.  Given  4x*—2xy  =  12, 

2y2-{-3xy  =  8,  to  find  a;  and  y. 
Assume  a;  =  zy,  i.  e.  substitute  zy  for  x. 

Ans.  x  =  ±2,ot  ±fv/7. 
y=±l,<V:Ffs/7. 

4.  Given  3x2+a?y=  68, 

fy*-\-3xy=  160,  to  find  a;  and  y. 

^4«5.  x=±±,  or  q:?y4y3, 

<7.)   Frequently,  by  a  little  reduction,  the  form  of  the 
equations  can  be  changed,  so  as  to  be  conveniently  solved. 

5.  Given  x"y — y  =z  21 

x^y — xy=  6,  to  find  x  and  y. 
rinding  y  from  each,  and  equating  the  two  values,  the  two  sides 
of  the  equations  will  be  found  to  have  a  common  factor. 

6.  Given  x--{-3x-\-y  =  73— 2xy, 

y--\-3y-\-x  —  44,  to  find  x  and  y. 
If  we  add  these  equations  and  transpose,  there  will  result  an  equa- 
tion, from  which  x+y  can  be  found. 


CHAPTER  VIII. 


RATIO  AND  PROPORTION. 


§  229.  In  considering  the  relative  magnitude  of  quanti- 
ties of  the  same  kind,  we  may  inquire,  either  how  much  one 
exceeds  the  other,  or  hoio  many  times  the  one  contains  the 
other.  The  former  of  these  relations  is  simply  the  differ- 
ence of"  the  quantities;  the  latter,  their  quotient,  is  also 
called  their  ratio.* 

§  230.   The  ratio  of  two  quantities  is  the  relation 

expressed  by  dividing  one  of  the  quantities  by  the 

other. 

2 
Thus,  the  ratio  of  2  to  3  is  -  (otherwise  written  2:3); 

that  of  a  to  b  is  y  (otherwise,  a  :  b). 

a.)  These  are  merely  different  ways  of  expressing  the 
same  thing ;  a  ratio  being  simply  a  fraction. 

b.)  The  first  term  (§111.  N.)  of  a  ratio  is  called  the  an- 
tecedent, and  the  second  the  consequent,  of  the  ratio. 

c.)  A  ratio  being  simply  a  fraction,  its  terms  may  be  both 
multiplied  or  both  divided  by  the  same  number  without  al- 
tering the  value  of  the  ratio  (§  113.  3).  Thus,  the  ratio  of 
2  to  3  is  the  same  as  that  of  2x5  to  3X5,  or  of  f  to  f .  So 
the  ratio  of  a  to  b  is  the  same  as  that  of  am  to  bm,  or  of 

a         b 

—  to  — .     That  is, 
m        m 


(ft)  Lat.,  relation. 


176 


RATIO  AND  PROPORTION. 

[§  231,  232. 

a 

2  2X5       f     a      am      m 

3  —  3X5  ~~f '  b  ~~bm  ""  6' 

m 

or,  2  :  3  =  2X5  :  3X5  =  |:  |,  &c. 

§  231.  A  proportion  is  an  equation  consisting  of 
two  equal  ratios. 

Thus  2  :  3  =  G  :  9,  or  §  =  -g-  is  a  proportion. 

~  7  am 

bo,  a  :  6  =  ?n  :  ra,  or  r  =  — . 

b       n 

Note.  Instead  of  the  sign  of  equality,  four  dots  (::)  are  some- 
times used.  Thus,  we  may  write  indifferently  2  :  3  :  :  6  :  9,  or 
2  :  3  =  6  :  9  (read  in  either  case  2  is  to  3  as  6  to  9).  The  sign  of 
equality  is,  however,  preferahle. 

a.)  The  terms  of  the  ratio  are  called  also  terras  of  the 
proportion.  The  first  and  last  terms  are  called  the  ex- 
tremes, and  the  second  and  third,  the  means  of  the  propor- 
tion. 

b.)  If  the  second  and  third  terms  be  the  same,  that/man- 
tity  is  said  to  be  a  mean  proportional  between  the  other 
two.     Thus, 

2:4  =  4:8;     a2  :  ab  =  ab:b*; 
o  _  4        a*___ab 
or  *~~*;     ab~W 

Here  4  is  a  mean  proportional  between  2  and  8 ;  and 
ab,  between  a2  and  b-. 

Note.  In  this  case,  the  three  terms  are  said  to  he  in  continued 
proportion. 

§232.    Leta:b  =  k  :l,ovy  =  y. 

b       I 

Clearing  of  fractions,  alz=bk.     That  is, 

In  any  proportion,  the  product  of  the  extremes  is  equal  to 

the  product  of  the  means. 

Thus,  if  5  :  7  =  10  :  14,  then  5x14=  10x7. 
Hence, 


§233.]  MEAN  PROPORTIONAL. — EQUAL  PRODUCTS.   177 

a.  Cor.  i.)  If  any  three  terms  of  a  proportion  be  given, 
the  fourth  may  be  found. 

For,  if  the  means  and  one  extreme  be  given,  the  other 
extreme  will  be  found  by  dividing  the  product  of  the 
means  by  the  given  extreme.  Or,  if  the  extremes  and  one 
mean  be  given,  the  other  mean  will  be  found  by  dividing 
the  product  of  the  extremes  by  the  given  mean.  Thus, 
If  x  :  6  =  11  :  22,  then  22a;  =66;  and  x  =  3. 

So,  if        5  :  x  —  10  :  40,  then  10a;  =  200  ;  and  x  =  20. 
Or,  if      5  :  13  =  15  :  x,  then  x  =  39. 

Note.    The  last  is  the  form  ordinarily  used  in  Arithmetic. 

b.)  Again,  let  a  :  x  =  x  :  b. 

Then  ab=zx2.     Hence, 

Cor.  II.  It  three  terms  be  in  continued  (§  231.  b.  N.)  pro- 
portion, the  product  of  the  extremes  is  equal  to  the  square  of 
the  mean. 

Thus,  if    2:  12  =  12:72,  then  2X72  =  12X12  =  129. 

c.)  Also,  if  a  :  x  =  x  :  b,  then 

i 
a;2  =  a&;  and  x=(ab)~.     Hence, 

Cor.  in.  The  mean  proportioned  betioeen  two  numbers  is 
equal  to  the  square  root  of  their  product. 

Thus  if  3  :  x  =  x  :  48,  then  x  =  (3X48)^  =  12. 

Find  a  mean  proportional  between  1  and  9  ;  between  2 
and  8 ;  between  5  and  500 ;  between  a2  and  62  ;  between 
R-\-x  and  R — x. 

§233.   Let  al  =  bL 

Dividing  both  members  by  b  and  by  7, 

t—ti  or  a  :  b  =  k":  I.    Hence, 
b       I' 

If  the  product  of  two  numbers  be  equal  to  the  product  of 

two  other  numbers,^  two  factors  of  either  product  may  be 

made  the  means,  and  the  two  factors  of  the  other  product  the 

extremes  of  a  proportion. 


178  RATIO  AND  PROPORTION'.  [§  284. 

Thus,  if  7X15  =  3X35, 

then         7  :  3  =  35  :  15,   or  7  :  35  =  3  :  15,  &c. 

So,  if     x"x— x"  2 [=  x"  (x—x") ]  =  A*—  x"  2, 
then  x"  :  A— x"  —  A\-x"  :  x— x" . 

1.  What  proportion  results  from  the  equation  &m(a-\-b) 
sin  (a— b)  =  sin2 a— sin25? 

Ans.  sin  (a — b)  :  sin  a — sin  Z>  =  sin  a-\-e'm  b  :  sin(a-\-b). 

2.  What  proportion  from  the  equation  sin  b  sin  G  = 
sin  c  sin  B? 

a.)  Also,  if  x2  =  a  5,  then  a  :  a;  =  a:  :  b. 

Hence,  evidently, 

Cor.  If  the  'product  of  two  numbers  be  equal  to  the 
square  of  a  third,  this  last  is  a  mean  proportional  behceen 
the  other  two. 

Thus,  if    122  =  2x72,  then  2:  12  =  12  :  72. 
So,  if        y2  =  R2 — x-,  then  R-\-x  :  y  =  y  :  R — x. 

Transform  the  following  equations  into  proportions. 

1.  y2  =  2Rx — x2.  Ans.  x  :  y  —  y  '•  2R — x. 

2.  y2  =  2px.  Ans.  x  :  y  =  y  :  2p. 

3.  R2  =  tan  a  cot  a  ;  A2  =x"x. 

§  234.    Let  a:b  — 7c:  I,  or  y  =  t- 

b       I 

I.   Multiplying  by  b,  and  dividing  by  Jc, 

a      b  7       7/ 

-  =  -  ;  or  a  :  «  =  b  :  I. 

Mis  C 

Or,  multiplying  by  /,  and  dividing  by  a, 

-  =- ;   or  I :  b-=  k  :  a.     Hence, 
b      a 

The  means  or  the  extremes  of  a  proportion  may  exchange 

places. 

Thus,  if      2  :  3  =  8:  12,  then  2  :  8  =  3  :  12. 

Note.    The  interchange  of  the  means  is  called  alternation'; 

(I)   Lat.  alterno,  to  interchange ;   hence  alternando,  by  inter- 
changing. 


§235,  236.]  inversion. — composition. — division.   179 

and  the  quantities  are  said  to  be  in  proportion  alternately,  or  alier- 
nando. 

a  Jc 

§235.   II.    Again  1-~  =  1-^-  -. 

—  —  —  ;  or  b  :  a  =  l :  Jc.     Hence, 
a      k 

The  terms  of  each  ratio  of  a  proportion  may  exchange 

places  ;  i.  e.  the  antecedent  may  be  made  consequent,  and 

the  consequent,  antecedent. 

Thus,  if       2  :  3  =  8  :  12,  then  3  :  2  =  12  :  8. 

Note.    This  is  called  inversions  ;  and  the  quantities  are  said 
te  be  in  proportion  by  inversion,  or  inveriendo. 

§236.   III.   Adding  ±1  to  each  side, 

.-.  (§114.  a)  C~-  =-T~>  or  a±b  :  b  =  Jc±l:  I.  (1) 

b      I  b  I 

Again  8  235)    -=-.     .M±:=l±j. 

a±b      Jc±l  , 7  ,,77 

=  — — ;  or  a±o  :  a  =  k±l  :  Jc.  (2 

a  k 

Hence, 

The  sum  or  difference  of  the  first  and  second  is  to  either 
the  first  or  second,  as  the  sum  or  difference  of  the  tJd-d  an! 
fourth  is  to  the  third  or  fourth. 

Thus,  if    7:5  =  14:  10,  then  7±5  :  7  =  14±10  :  14. 

Note.  In  this  case  the  quantities  are  said  to  be  in  proportion  by 
composition,"  or  componendo,  when  the  sum  is  taken;  and  by  divis- 
ion or  dividendo0,  when  the  difference  is  taken. 

a.)    Also  a-\-b  :  k-\-l—  a  :  Jc;  §  234. 

and  a — b  :  k — I  =  a  :  h. 

a+b  :  k-\-l  =  a—b  :  k—l. 

or  (§  234)  a-\-b  :  a—b  =  fc+Z :  k—l. 

(?ft)  Lat.  inverto,  to  invert;  hence  invertendo,  by  inverting,  (n) 
Lat.  compono,  to  compound ,  hence  componendo,  by  compounding. 
(o)  Lat.,  from  divido,  io  separate ;  by  separating. 


180  RATIO  AND  PROPORTION.  [§  237,  233. 

Hence, 

Cor.    The  sum  of  the  first  and  second  is  to  their  differ- 
ence, as  the  sum  of  the  third  and  fourth  is  to  their  difference. 
Thus   3:2  =  6:4;    .-.  3+2  :  3—2  =  6+4  :  6—4. 
Hence  (f§  234-236), 

§  237.  If  four  quantities  he  in  proportion,  they  will  be 
in  proportion  by  alternation,  by  inversion,  by  composition, 
or  by  division. 

§  238.    Let  a  :  b  =  k  :  I,  or  %  =  T. 

b      I 

Adding  ±n  (§  42.  a), 

a  ,  k  ,  a±nb       k±nl 

a±nb  :  b  =  k±nl :  I.  (1) 

Again  (§  235),  -  =  t  ',  and  -  ±m  =  -±m; 

Ct       /C  (X  fc 

b±ma      l±mk 

or  =  —j—. 

a  k 

b±ma  :  a  =  l±mk  :  k.  (2) 

We  have  also  (§  234)  a:k  =  b:l; 
and  from  (1),     a±?ib  :  k±nl=  b  :  l=za  :  k; 
and  from  (2),    b±ma  :  l±mk  =z  a  :  k  =  b  :  I. 
.'.  (§  231)  a±nb  :  k±nl=b±ma  :  l±mk.  (3) 

Now  (§  230.  c)  ma  and  wft  have  the  same  ratio  as  a  and 
k ;  also  w6  and  nl,  the  same  as  b  and  /.     Hence, 

Jf  either  both  antecedents  or  both  consequents  be  increased 
or  diminished  by  quantities  having  the  same  ratio  as  either 
consequents  or  antecedents,  the  results  will  be  in  proportion 
with  either  the  antecedents  or  consequents,  or  with  each  other. 

Thus,  if  2  :  4  =  6  :  12 ;  then  2±3  :  4  =  6±9  :  12  ; 
and     2  :  4±1  =  6  :  12±3 ;  2±3  :  4±1  =  6±9  :  12±3. 

Notes.  (1.)  ma  and  mk  are  called  equimultiples**  (i.  e. 
products  by  a  common  multiplier)  of  a  and  k.     (2.)  If  m  and  n  be 

(p)   Lat.  a;quus,  equal,  and  multiplico,  to  multiply  (§66.  Note 


§  239-241.]    EQUIMULTIPLES. — SUMS. — POWERS.  181 

each  equal  to  unity,  the  formula  (1)  and  (2)  of  this  section  become 
identical  with  (1)  and  (2)  of  §236. 

§  239.    Let  a  :  b  =  k  :  I. 

ma      mk      ra        k  ,  ma      mk 

The"       T=T;    ni,  =  M-^nb=nV      H2'  ''  * 

ma  :  b  =  mk  :l;     a  :  nb  —  k  :  nl; 
and  wzrc  :  nb  =  m&  :  n?.     Hence, 

Equimultiples  of  the  antecedents  and  of  the  consequents 
of  a  proportion  will  be  in  proportion,  either  with  the  origi- 
nal antecedents,  or  consequents,  or  with  each  other. 
Thus,  if  2:4  =  6:12;  then  2Xo  :  4X7  =  6X0  :  12X7. 
Or,         2X5:6X5  =  4X7:12X7  =  4:12  =  2:6. 

Note.  We  may,  obviously,  multiply  both  terms  of  a  ratio  (§230. 
c)  or  both  the  antecedents,  or  consequents  (§42.  c,  d)  of  Ta  propor- 
tion, by  a  common  multiplier,  without  destroying  the  proportionality. 

§240.   Let     a  :  b  =  e  :f=g  :  h  =  k  :l. 
Then  ab  =  ab;    and  (§  232)  af=  be;    ah  =  bg ;    al  =  bk. 

a(b+f+h+l)  =  b(a+e+g+k). 
.-.  (§  233)  a-\-e-\-g+k :  b-\-f-\-h-\-l  z=a:b  =  e:f,&c.  Hence, 

In  any  number  of  equal  ratios,  the  stem  of  all  the  antece- 
dents is  to  the  sum  of  all  the  consequents  as  any  one  of  the 
antecedents  is  to  its  consequent. 
Thus,  if  1:2  =  3:6  =  4:8  =  5:10, 

then  1+3+4-1-5  :  2+0+8+10  =  1:2. 

§  241.    Let  a  :  b  =  k  :  I. 

Then  (§  52.  N.)  ~  =  ^- ;   or  a\  :  bn  —  k\  :  l\     Hence, 

Like  poxoers  of  proportional  quantities  are  proportional. 

Thus,  if  1:4  =  64:256, 

then  I3  :  43  =  643  :  2563; 

and  71:74  =  ^4:  ^256. 

Note.    The  ratio  of  the  squares  of  two  quantities  was  formerly 
called  the  duplicate  ;  that  of  the  cubes,  the  triplicate ;  of  the  square 
ALG.  16 


182  RATIO  AND  PROPORTION.  [§  242-244. 

a:id  cube  roots,  the  subduplicate  and  subtriplicatc,  ratio  of  the  quan- 
tities themselves.  The  ratio  of  the  square  roots  of  the  cubes  (i.  e.  of 
the  three  half  powers)  is  sometimes  called  the  sesquiplicate  ratio  of 
the  quantities. 

§  242.  Let  a  :b  —  k  :  I;   e  :f=g  :  h ;  and  r  :  s  =  x :  y. 

ml  a    e      r       k     q     x         aer      kqx 

Then  -  .  -  .  -  =  -  .  |  .  _  ;  or  —  =  -f- ; 

u    f     s        Ihy         bfs      Ihy 

or  aer  :  bfs  =  hgx  :  Ihy. 

The  same  will  evidently  hold  of  any  number  of  propor- 
tions.    Hence, 

The  products  of  the  corresponding  terms  of  any  number 
of  proportions  are  proportional. 
Thus,  if        i  :  3  =  6  :  18,  and  10  :  6  =  15  :  9, 

then  1X10:3X6  =  6X15:18X9. 

Notes.  (1.)  When  the  terms  of  two  ratios  are  thus  multiplied 
together,  the  ratios  are  said  to  be  compounded.  (2.)  If  equal  ra- 
tios are  compounded,  we  obtain  the  ratio  of  the  powers  of  the  quan- 
tities (§241). 

§  243.  The  following  exhibits,  very  briefly,  most  of  the 
principles  above  demonstrated  (§§  232-242).  If  the  truth 
of  any  of  these  expressions  is  not  self-evident,  write  the  ra- 
tios in  the  form  of  fractions. 

1.  ar:a  =  br:  b  ;  or  —  =  ^(§§113.  1;  114;  230.  a). 

a        b 

2.  abr  —  abr.     §232. 

3.  ar:br  =  a:b.     §234.     See  113.  3. 

4.  a  :  arz=b  :  br.     §  235. 

5.  ar±a  :  a  =  br±b  :  b  ;  or  a  (V±l)  :  a  =  b  (r±l)  :  b. 
§236. 

6.  a  (r+1)  :  a  (r— 1)  =  b  (r+1)  :  b  (r—  1).    §  236.  Cor. 
Note.    Other  principles  may  be  exhibited  in  like  manner. 

§  244.  When  the  first  of  four  quantities  is  to  the  second 
a3  the  fourth  is  to  the  third  (i.  e.  as  the  reciprocal  (§  18)  of 
the  third  is  to  the  reciprocal  of  the  fourth),  they  are  said  to 
be  inversely  (§  235)  or  reciprocally  proportional. 


§  245,  246.]  variation.  183 

Thus,  if,  on  a  railroad,  a  freight  train  runs  15,  and  a  pas- 
senger train  30  miles  an  hour,  their  times  of  passing  over 
equal  distances  on  the  road  will  be  inversely  or  reciprocal- 
ly proportional  to  their  velocities.     That  is, 

Time  of  1st  :  Time  of  2d  =  Vel.  of  2d  :  Vel.  of  1st  = 

ovT:  T'=  V  :   V—  ■* 


Vel.  of  1st,  '  Vel.  of  2d. '  ut  "  '  ^    ~        •       —  y  •  y" 
If,  however,  they  run  equal  times,  as  3  hours,  then  the 
distances  will  be  directly  proportional  to  their  relocitiee. 

VARIATION. 

§  245.  These  relations  are  sometimes  concisely  express- 
ed by  saying,  that  one  class  of  quantities,  or,  still  more 
concisely,  that  one  quantity  varies  directly  or  inversely 
as  another.  This  form  of  expression  is  denoted  by  this 
symbol  go,  or  ==,  placed  between  the  quantities.  Thug, 
x&y,  or  x  =  y,  (read  x  varies  as  y). 

Thus,  in  the  examples  of  the  last  section,  the  time  is  said 
to  vary  (or  to  be)  inversely  or  reciprocally,  and  the  distance 

directly,  as  the  velocity.     Or,  I7  go  — ;  D  go  V. 

So,  the  number  of  men  required  to  accomplish  a  work  in 
a  given  time  varies  directly  as  the  amount  of  work ;  if  the 
amount  of  work  be  given,  the  number  of  men  varies  in- 
versely as  the  time  allowed. 

§  246.  If  x==y,  then  we  shall  have,  obviously, 
x  :  x1  =  y  :  y>  •   ov  x  :y  =  x'  :y'. 

x      x' 

-  —  -,  =  m,  a  constant  numbei*.        (1 ) 

Also,  x  =z  my ;  and  y  =  — x.  (2) 

m 
Hence, 

When  one  quantity  varies  directly  as  another,  (1.)  the 
ratio  of  the  numbers  by  which  they  are  expressed  is  con- 
stant; and  (2.)  each  is  equal  to  the  other  multiplied  by 
some  constant  number. 


184  EATIO  AND  PROPORTION.  [§  247. 

§247.    Let  a: go -.     Then  x  :  xfz=-  :  -  =  */:  y. 

y  y  y'    y  u 

xy  =.  x'y'  =:  m,  a  constant  number.  (1) 

• ,  otto 

Also,  3?  = —  ;andy  =  ^-.  (2) 

y  a      x  v 

Hence, 

If  one  quantity  varies  reciprocally  as  another,  (1.)  the 
product  of  the  numbers  by  which  they  are  expressed  is 
constant ;  and  (2.)  each  is  equal  to  a  constant  quantity  di- 
vided by  the  other. 

a.)  The  converse  of  the  principles  in  this  and  the  last 
section  is  evidently  true. 

Hence,  (3.)  any  equation,  containing  variable  quantities, 
way  be  written  as  an  expression  of  variation ;  and  may  be 
simplified  by  dropping  any  constant  factor  on  either  side. 

Also,  (4)  if  all  the  factors  on  one  side  be  constant  the 
other  side  is  constant  (§§  246.  1 ;  247.  1). 

Thus,  if  we  have  the  area  of  a  circle  —tiR'2,  n  being 
constant9,  then  the  area  varies  as  the  square  of  the  radius ; 
or  area  =  2i2. 

So,  S  representing  the  space  fallen  through  by  a  falling 
body,  and  T,  the  time  of  its  descent,  i£  S ~  mT2,  m  being 
constant,  then  the  space  varies  as  the  square  of  the  time ; 
or  S  =  T2. 

Again,  if  the  area  (A2)  of  a  rectangle  —  its  base  (x)  X 

its  altitude  (y)  ;  i.  e.  if  A2  =  xy,  then 

..     ,  the  area      .  A2  1  1 

the  base  =  -; — — — ;  or  x  =  —  =  A2- ;  and  x  w  - ;  or 

the  altitude    *  y  y  y 

the  base  varies  inversely  as  the  altitude. 

b.)  In  the  last  example,  the  area  varies  as  the  product  of 
the  base  and  altitude.  So  the  solidity  of  a  parallelopipe- 
don  varies  a3  the  product  of  its  length,  breadth  and  thick- 
ness. 

(q)  it,  Greek  letter  pi,  Eng.  p  ;  the  initial  (§  1.  d)  of  Trepujtipeta, 
periphery,  circumference.  In  common  use,  7r  =  3.14159  &c  the 
circumference  of  the  circle  whose  diameter  is  unity. 


§  248.]  VARIATION. — PROBLEMS.  185 

c.)  I£xzc-,  then  x  varies  directly  as  y,  and  inversely  as 
z.  Thus,  the  weight  ( W)  of  a  body  above  the  surface  of 
the  earth  varies  directly  as  its  mass  {M),  and  inversely  as 
the  square  of  its  distance  [D)  from  the  centre  of  the  earth. 

rrr  M 

That  is,  W  co  -r-^. 

JJ~ 

§  248.  1.  If,  above  the  surface  of  the  earth,  the  weight 
of  a  given  body  (i.  e.  its  gravitation  towards  the"  earth)  va- 
ries inversely  as  the  square  of  its  distance  from  the  centre 
of  the  earth,  how  high  must  the  body  be  raised,  that  its 
weight  may  be  only  half  what  it  was  at  the  surface  ? 

Let  x  r=  the  height  above  the  surface  of  the  earth ; 
r  =  the  radius  of  the  earth ;  and 

w  ■==.  the  weight  of  the  body  at  the  surface. 

1  1 

Then  w  :  %w  —  —  :  -—. — —  =  (r-\-z)  2  :  r2  ; 
r-     (r-f-x)- 

or  1  :  l=(r-\-x)*  :  r2. 

i(r-\-x)  2  =  r2;  or  xn--\-2rx  =  r2. 

x  =  — r±r^/2. 

Note.  Taking  the  upper  sign,  and  finding  ^y2  approximately, 
we  have  x  —  JtJULr.  The  lower  sign  gives  the  distance,  measured 
downward  (§5)  through  the  centre. 

2.  How  far  must  the  body  be  removed  from  the  surface, 
that  its  weight  may  be  w'  ? 

Here  we  have     w  :  w' '=  (f-\-x) 2  :  r- ; 

or  «/w  :  */io'  =  r-\-x  :  r 

w 
x  —  — r±rJy—r. 

w' 

3.  How  much  weight  will  the  body  lose,  if  it  be  remov- 
ed a  given  distance  (D)  from  the  surface  ?  and  what  will 
be  its  weight  there  ? 

Here  .  w  :  w>  =  (r-f-Z>)  2  :  r2 

w  :  w-w'  —  (r+D)  2  :  (r+D)  2-r2.         §  236. 
*16 


186  EQUIDIFFERENT  SBRIES.  [§  249. 

,       (2rZ>4-D2)w     ,     ,         „      .  , 
w-w>  =  ^2+2rjD+ja.  tlie  loss  of  weight ; 

r* 

(r+X»)2 

If  D  is  very  small  compared  with  r,  D^  may  be  neglected,  and 
we  shall  have 

2D 

w — w'  =  — — — -w. 
r-\-2D 

Let  D  =  l,  2,  5,  10,  100,  1000  miles,  w=l  pound,  and  r  — 
4000  miles ;  and  find  the  values  of  w'  and  w — iv'. 


CHAPTER  IX. 


EQUIDIFFERENT,  EQUIMULTIPLE 
AND  HARMONIC  SERIES. 


I.  EQUIDIFFERENT  SERIES. 

§  249.  A  series  of  quantities  such  that  each  differs 
from  the  preceding  by  a  constant  quantity,  is  called 
an  equidifferent  series ;  and  sometimes  an  arith- 
metical series  or  progression. 

a.)  Such  a  series  can,  of  course,  be  continued  to  any  ex- 
tent ;  and  its  character  is  determined,  if  we  know  any  one 
of  its  terms  and  their  common  difference. 

Thus,  if  7  be  one  of  the  terms,  and  3  the  common  differ- 
ence, we  shall  have  the  series, 

....    —5,  —2,  1,  4,  7,  10,  13,  16,     ...     . 

Or,  if  8  be  one  of  the  terms,  and  — 2  the  difference,  we 
shall  have 

....    12,  10,  8,  G,  4,  2,  0,  —2,  —  4,    .    .    .    . 


§250,251.]  LAST  TERM. — SUM.  187 

b.)  If  the  common  difference  be  positive,  the  series  is 
called  increasing ;  if  negative,  decreasing.  The  first  of  the 
series  in  a  above,  is  an  increasing  series ;  the  second,  a  de- 
creasing series. 

c.)  Though  every  series  may  be  continued  without  limit, 
we  ordinal  ily  have  occasion  to  consider  only  some  definite 
number  of  terms,  of  which  the  two  extremes  are  called  the 
first  and  last  terms. 

§  250.  If  a  be  the  first,  and  I  the  last  of  n  terms  of  an 
equidifterent  series,  and  D  their  common  difference,  we 
shall  have 

1st,     2d,  3d,  (n-l)th,  nth, 

a,   a-\-D,   a-\-2D,     .     .     a-\-(n— 2)D,   a-\-(n—\)D  or  /; 

whence,  obviously,       l=z  a-\-(n — 1)D.  (1) 

That  is, 

The  last  term  is  equal  to  the  first  term,  plus  the  product 

of  the  common  difference  by  the  number  of  terms  less  one. 

Note.  Of  course,  the  common  difference  must  be  taken  positive 
or  negative,  according  as  the  series  is  increasing  or  decreasing. 

1.  What  is  the  7th  term  of  the  series  1,  3,  5,  &c.  ? 
Here  a  =  1,  D  —  2,  and  n  —  7. 

l  =  a+(n—l)D=l-\-Gx2  =  13 

2.  Given  a  =  25,  D  =  —2,  and  n  =  14 ;  to  find  I. 

Ans,  — 1. 

3.  Given  a  ■==.  0,  D  —  1,  and  n  =  100  ;  to  find  I. 

Ans.  99. 

§  251.  If  s  represent  the  sum  of  n  terms  of  a  series,  we 
shall  have 

s  =  aJr(a+B)+(a+2I))  .  .  +{  [a-f-0-l)Z>](  =  Z)  y  5 
and,  writing  the  terms  in  the  reverse  order,  obviously 
s  =  l+(l— D)+(l— 2D)      .    +{  [?—  (n— l)Z>)](  =  a)  }. 
/.  Adding  the  equations, 
2s=(a+l)+(a+l)-\-(a+l)      .     .      +(a+l)  =  n  («+?). 


188  EQUIDIFFERENT  SERIES.  [§  252. 

s=9^pl.  (2)  That  is, 

The  sum  of  any  number  of  terms  of  an  equidifferent  se- 
ries is  equal  to  the  number  of  terms  into  half  the  sum  of  the 
extremes. 

1.  What  is  the  sum  of  20  terms  of  the  series  1,  3,  5,  7, 
&c? 

Here  a  =  l,  D=z2,  and  n  =  20. 

l=a-\-(n—l)Z>  =  1+19X2  =  39. 
s  —  bi(a+l)  =  \  .  20(1+39)  =  400. 

2.  Given  a  ■=  1,  D=  1,  and  n  =.  10  ;  to  find  £  and  s. 

Arts.  1=  10,  s  ==  55. 

3.  Given  a  =  20,  Z>  = — 2,  and  «  =  21 ;  to  find  ?  and  s. 

Ans.  I  =  —20,  s  =  6. 

4.  Let  a  =  20,  D  =  — 2,  and  w  =  11 ;  and  find  I  and  5. 

§  252.  a.)  It  is  obvious  from  the  addition  of  the  two 
series  above  (§  251),  that  the  sum  of  any  two  terms  equidis- 
tant from  the  extremes  is  equal  to  the  sum  of  the  ext)  ernes. 

Or,  beginning  with  a,   the  mth  term  =  «+(?» — \)D; 
and,  beginning  with  I,       the  mtla.  term  =  / — (m — 1)D. 

Now  the  sum  of  these  two  terms,  equidistant  from  the 
extremes  is  a-\-l. 

b.)  Hence,  if  the  number  of  terms  be  odd,  the  middle 
term  is  half  the  sum  of  the  extremes. 

c.)  Such  a  term  is  called  an  equidifferent  mean,  and 
sometimes  an  arithmetical  mean. 

d.)  The  equidifferent  mean  between  two  quantities  is 
found  by  taking  half  their  sum.  Thus,  the  equidifferent 
mean  between  1  and  2  is  11-,  or  1.5 ;  between  1  and  1.5, 
1.25;  between  5  and  15,  10. 

e.)  The  middle  term  is  also  equal  to  the  sum  of  all  the 
terms  divided  by  their  number.     For 

5=  \{a-\-l)n ;  .'.  -=  \{a-\-l)  =  the  middle  term. 

Tit 


§  253-255.]       FORMULAE. — INTERPOLATION.  189 

Note.  A  mean  of  several  quantities,  whether  they  be  equidiffe- 
rent  or  not,  is  found  by  dividing  the  sum  of  the  quantities  by  their 
number.  The  mean  or  average  temperature  for  a  week  or  month  is 
found  in  this  way  from  the  several  daily  temperatures  observed  dur- 
ing the  given  period. 

§  253.  /.)  If  (§§  250,  251)  we  substitute  in  (2)  the  value 
of  /  in  (l),'we  shall  have  s  in  terms  of  a,  D  and  n.     Thus, 
s  =  nh(a+l)  —  na-\-hi(n—l)D  =  w[a-{-£(w— 1)2>]. 

§  254.  The  formula?,  I  ==■  a-\-(ii — 1)Z>,  and  s  =  fyi(a-\-l), 
should  be  carefully  remembered.  They  contain,  it  will  be 
observed,  five  quantities.  If  any  three  of  these  be  given, 
we  shall  have  two  equations  containing  two  unknown  quan- 
tities which  may  therefore  be  determined  (§§  124-128). 


a.)  In  fact,  from  the  first,      a  =  I — (n — 1)D ; 

(3) 

n — 1 

(4) 

a                              »=£+i. 

(5) 

b.)  In  like  manner,  from  the  second, 

2s 
a-\-l 

(6) 

2s 
a  = 1 ; 

n 

(7) 

n 

(8) 

§  255.  c.)  From  formula  (4)  we  can  interpolate*  any 
number  of  equidifferent  means  between  two  given  extremes. 
For  let  it  be  required  to  interpolate  m  intermediate  terms 
between  a  and  b.  We  shall  have  the  whole  number  of 
terms,  n,  equal  to  m-\-2.     Hence,  n — 1  =  m-j-1,  and 

_  /__  I — a  \ b — a 

\n~—\)~m+l\ 
Hence  we  have  the  series," 

(r)  Lat.  iaterpolo,  Fr.  interpoler,  to  insert. 


190  EQUIMULTIPLE  SERIES.  [§256,257. 

a,  a-\ — ,    a-\-2 — — ,     .     .        a4-(m4-l) — —  (=b). 

m-\-Y        '     m-f-1  '  v     '     Jm-\-\K       ' 

1.  Interpolate  8  equidifferent  means  between  1  and  10. 

2.  Find  6  equidifferent  means  between  1  and  15. 

§256.   1.    Given  a  =  1,  D  =  l,  and  »  =  100;   to  find  7 
and  s.  Ans.  1=100,  s  =  5050. 

2.  What  is  the  rath  term  of  the  series  of  example  1  (i.  e. 
the  rath  term  of  the  natural  series  1,  2,  3,  4,  &c.)  ? 

Ans.  n. 

3.  What  is  the  sum  of  n  terms  of  the  series  1,  2,  3,  &c.  ? 

Ans.  ^±L}. 

4.  What  is  the  nth.  term  of  the  series  1,  3,  5,  7,  &c.  ? 

.Jras.  2rc — 1. 

Substitute  for  n,  1,  2,  3,  4,  5,  &c. 

5.  What  is  the  sum  of  n  terms  of  the  above  series  of 
odd  numbers,  1,  3,  5,  &c.  ?  Ans.  n2. 

Substitute  for  n  as  above. 

6.  Suppose  a  body,  falling  freely  to  the  earth,  descends 
m  feet  the  first  second,  dm  the  second'second,  bin  the  third, 
&c.  Now  if  its  fall  occupy  T  seconds,  how  far  will  it  fall 
in  the  last  second?  Ans..  (2T—V)m. 

7.  How  far  will  it  fall  in  the  whole  T  seconds  ?  i.  e. 
what  is  the  sum  of  the  series,  m,  om,  5m,  &c,  to  T  terms  ? 

Ans.  m  T2. 
Substitute,  in  these  two  examples,  for  T,  5,  6,  7,  8,  10,  &c.     Al- 
so find  the  value  of  the  expressions  thus  obtained,  on  the  supposition 
tbat  m  —  16^. 

II.  EQUIMULTIPLE  SERIES. 

§  2-57.  A  series,  such  that  each  term  is  formed  by 
multiplying  the  term  immediately  preceding  by  a  con- 
stant multiplier,  is  called  an  equimultiple  series ; 
sometimes  also  a  geometrical  series  or  progression, 
or  a  progression  by  quotient. 


§258.]  LAST  TERM.  191 

Note.  The  constant  multiplier  has  been  sometimes  called  the 
ratio.  For  convenience  and  distinctness,  however,  we  shall  call  it 
the  common  multiplier,  or  simply  the  multiplier. 

a.)  Such  a  series  can,  of  course,  be  continued  to  any  ex- 
tent ;  and  its  character  is  determined,  if  we  know  any  one 
of  its  terms  and  the  common  multiplier. 

Thus,  if  7  be  one  of  the  terms,  and  3  the  common  multi- 
plier, we  shall  have  the  series, 

TJ7,         y,         6h,         i,         -1,         OO,  .... 

So,  if  8  be  one  of  the  terms,  and  |  the  multiplier,  we 
shall  have 

...     32,    1G,   8,    4,   2,    1,    i,   \,     .     .     . 

b.)  If  the  common  multiplier  be  greater  than  unity,  we 
shall  have  an  increasing  series ;  if  less,  a  decreasing  series. 
The  first  of  the  two  series  in  a,  above,  is  an  increasing,  the 
second  a  decreasing  series. 

c.)  Though  every  series  may  be  continued  without  limit, 
we  ordinarily  have  occasion  to  consider  only  some  definite 
number  of  terms,  of  which  the  two  extremes  are  called  flhe 
first  and  last  terms. 

§  258.   If  a  be  the  first,  and  I  the  last  of  n  terms  of  an 

equimultiple  series,  and  m  the  common  multiplier,  we  shall 

have 

1st,    2d,       3d,       4th,       5th,  (?i— l)th,  Tith, 

a,    am,   am2,    amz,    am*,     .     .     .      amn~2,   am"'1  or/. 
Whence,  obviously,         /  — amn_1.  (1) 

That  is,  to  find  the  nth  term  of  an  equimultiple  series, 
Multiply  the  first  term  by  the  (n — \)th  power  of  the  com- 

mon  multiplier. 

1.  What  is  the  6th  term  of  the  series  1,  2,  4,  &c  ? 
Here  a  =z  1,  m  =  2,  and  n  z=  6. 

l(=  am"-1)  =  1x2  5  =  32. 

2.  Given  a  =  3,  m  =  2,  and  n  =  10 ;  to  find  /. 

Ans.  1536. 


192  EQUIMULTIPLE  SERIES.  [§258. 


8.    Given  a  =  64,  m  =  h,  and  n  =  8  ;  to  find  I. 

4.  Given  a  =  $100,  m  =  1.06,  and  n  =  10;  to  find/; 
i.  e.  to  what  will  $100  amount  in  10  years  at  6  per  cent, 
compound  interest?  Ans.  1=- $179.09. 

5.  "What  is  the  amount  (A)  of  p  dollars,  at  compound 
interest  for  t  years,  at  the  rate  r  ? 

Here  we  have 

1-J-r  =  the  amount  of  one  dollar  for  one  year. 
p(l-\-r)  =  "  p  dollars  " 

p(l-\-r)(l+r)  =  "      ^(1+r)     "  " 

&c. 

Or,         p  =  the  amount  at  the  beginning  of  the  jirst  year ; 
p(l-\-r)=  «  "  wconrf    « 

■  •  •  •  • 

jp(l-f-r)re-1=  "  «  rath  " 

p(l-j-ry=  «  «  (<+l)th  " 

i.  e.  at  the  ewe?  of  t  years. 

The  successive  amounts  constitute,  obviously,  an  equi- 
multiple series  ;  in  which  we  have  given  a=p,  m=-  1-J-r, 
and  n  =  t-\-\  ;  to  find  1=  A.  Ans.  A  =zp(l-\-r)(. 

G.  What  is  the  amount  of  $50  at  6  per  cent,  compound 
interest  for  12  years?  Ans.  $100.61. 

7.  What  sum,  at  the  rate  r,  will  amount  to  A  dollars  in 

t  years  ?  .  A  _       .     , 

J  Ans.  p  =  ■    ,    ,,.     See  4,  above. 

(i-K) 

8.  What  principal  at  6  per  cent  will  amount  to  $1000, 
in  10  years?  Ans.  $558.37. 

9.  At  what  rate  of  compound  interest  will  p  dollars 

amount  to  A  dollars  in  t  years ?  .  /A\L 

J  A?is.r=(  —  U— 1. 

Let  p  =  100,  A  =  150,  and  t  —  8 ;  &c. 

a.)    If  we  have  m  <  1,  and  n  —  &,  then  putting  m  =— y 

/IS 


§  259.]  LAST  TERM. — INFINITE  SERIES.  193 

(«t'  being  of  course  >  1),  we  shall  find       I  (  =  am"-1)  = 

/l  v00-!  a  am'      am!      rt#.  ,«0    o\«      >tm    ^  • 

«[— l       = = =  —  =0  (§  138.  3Y.     That  is, 

The  last  term  of  a  decreasing  infinite  equimultiple  se- 
ries is  zero. 

Notes.  (1.)  Of  an  infinite  series,  there  can  be  no  last  term. 
And,  on  the  other  hand,  in  forming  the  terms  by  multiplication,  we 
can  never  arrive  at  zero;  though  we  must  evidently  approximate  to 
it.  Hence,  the  inconsistency  in  speaking  of  the  last  term  of  an  in- 
finite aeries  is  compensated  by  placing  it  beyond  any  finite  number 
of  terms;  i.  e.  at  an  infinite  distance. 

(2.)   In  the  two  series, 

1>  b  h  h  &c-  5  and  2»  1>  b  h  &c-> 

any  term  whatever  of  the  first  is  half  the  corresponding  term  of  the 
second.  Hence,  the  last  terms  are  said  to  be  in  the  same  ratio. 
Now  this  comparison  can,  obviously,  be  made  only  between  terms  at 
some  definite  distance  from  the  beginning.  That  distance,  however, 
can  be  taken  as  great  as  we  please;  and  the  terms,  consequently, 
can  be  brought  as  near  zero  (and,  therefore,  as  near  equality)  as  we 
please,  while  the  ratio  remains  constant.  Thus  infinitesimals, 
though  regarded  as  equal  to  zero,  may,  like  finite  quantities,  have 
any  definite  ratio  to  each  other. 

§  259.  b.)  It  is  evident,  from  the  formation  of  the  sev- 
eral terms  of  an  equimultiple  series,  that  the  product  of  any 
two  terms  equidistant  from  the  extremes  must  be  equal  to 
the  product  of  the  extremes. 

In  fact,  if  a  be  the  first  of  n  terms,  the  term  which  has  p 
term3  before  it  will  be  am? ;  the  one  which  has  p  terms 
after  it,  being  the  (»—/>)  th,  will  be  equal  to  amn~p~'t. 

Hence  their  product 
ampXamn-p-i  —  a2mn-i  _  flXflmn-i  —  ai  (§  258). 

c.)  Or  again,  if  a  be  the  first  term,  and  m  the  multipli- 
er, we  shall  have  the  (jo-j-l)th  term  =  amP. 

(s)  It  is  evident  that,  since  m'>l,  a  finite  number  of  factors, 
each  equal  to  m',may  be  taken  sufficient  to  produce  any  finite  num- 
ber whatever.  Hence,  if  we  combine  an  infinite  number  of  these 
factors,  the  result  will  be  infinite. 

ALG.  17 


!94  EQUIMULTIPLE  SERIES.  [§260,261. 

But,  if  we  begin  with  I,  the  multiplier  is,  obviously,  —  ; 

m 

f  1  \p        I 
and  the  (»4-l)th  term  =  l( — ]  = — . 
v/        J  \m )        m" 

ampxl —  =  al. 
mp 

d.)  Hence,  if  the  number  of  terms  be  odd,  the  product  of 
the  extremes  tvill  be  equal  to  the  square  of  the  middle  term 
(it  being  equally  distant  from  the  two  extremes). 

e.)  Such  a  term  may  be  called  an  equimultiple  mean.  It 
is  sometimes  called  a  geometrical  mean,  and  is  simply  a 
mean  proportional  between  the  extremes  (§§  231.  b ;  232.  b). 

§  260.  If  s  represent  the  sum  of  n  terms  of  an  equimul- 
tiple series,  we  shall  have 

s  =.  a  -\-  am  -j-  am  -  —  am  3  -\-     .     .     -f-  amn~ l . 
Multiplying  by  m, 

ms  —  am  -\-  am  -  -\-  am3  -f-  am4  -f-     .     -f-  amn. 
Subtracting  the  first  of  these  equations  from  the  second, 
ms — s  z=z  anf — a ;  or  (in — l)s  =  a(m" — 1). 

^«K-=1).       (2)        '         That  is, 
m — 1 

To  find  the  sum  of  n  terms  of  an  equimultiple  series, 

Raise  the  midtiplier  to  the  nth  power,  and  subtraot  1  / 

multiply  the  remainder  by  the  first  term,  and  divide  the  pro 

duct  by  the  midtiplier  diminished  by  unity. 

ramn — a 

§  2GI.  a.)  We  have  s  = — ,  §  260. 

'  m — 1 

and  l  =  amn~K  §258. 

s_lrnr-a  That  is, 

m — 1 
To  find  the  sum  of  n  terms  of  an  equimultiple  series, 
Multiply  the  last  term  by  the  multiplier,  subtract  the  first 
term,  and  divide  by  the  midtiplier  less  one. 


§  261.]  SUM. — INFINITE  SERIES.  19"' 

k)   If  m  <  1,  both  mn—\  and  m—1  will  be  negative. 

In  that  case  it  is  convenient  to  change  the  signs  and  the 

order  of  the  terms,  thus  ; 

«(1— ran)  a—lm 

s  =  —^ ,  or  s  =  - . 

1—  m  1 — m 

1.  Find  the  sum  of  20  terms  of  the  series  1,  2,  4,  8,  &c. 
Here  a=.l,  m  —  2,  and  n  =  20  ; 

8=-(^-i)=i(^°-i)=1>048)WS, 

m — 1  2 — 1 

2.  Given  a  =  243,  m  =  |3  and  n  =  7 ;  to  find  Z  and  s. 

Ans.  ?=  i,  s  —  3641. 

3.  Given  a  =  1,  m  =  4,  and  n  =  5  ;  to  find  ?  and  s. 

Ans.  l=25Q;   s  =  341.     • 

4.  Given  arrl,m=  J,  and  ?z  =  6,  to  find  Z  and  s. 

M.11S.  I  ^^  J4TT  5    ^  ~~      23T3' 

c.)  If  m  <  1,  and  n  =  oo,  we  should  have,  reasoning  as 
in  §  258.  a,  am"  =  am™  =  0. 

s=    "  (4) 

1 — m 

That  is, 

The  sum  of  a  decreasing  infinite  equimultiple  series  is 
equal  to  the  first  term  divided  by  the  difference  between  uni- 
ty, and  the  common  multiplier. 

We  might  obtain  the  same  result  by  substituting  the 
value  of  I  (§  258.  a)  in  formula  (3)  of  §  261. 

Notes.  (1.)  If  n  is  infinite,  n — 1  is  infinite  also.  For,  if  n — 1 
were  finite,  n  being  greater  by  unity  than  a  finite  number,  must  be 
finite  also.  In  like  manner,  if  any  finite  quantity  whatever  be  sub- 
tracted from  infinity,  the  remainder  is  still  infinite.  (2.)  Hence, 
we  have  co=^r  oo±a.  That  is,  an  infinite  quantity  is  not  affected  by 
the  addition  or  subtraction  of  a  finite  quantity. 

1.  Given  a  =  1,  m  =  ^,  and  n  =  oo  ;  to  find  the  sum  of 
the  series.  Ans.  s  =  2. 

2.  What  is  the  sum  of  the  infinite  series,  whose  first 
term  is  1,  and  multiplier  ^  ?  Ans.  1£. 


196 


EQUIMULTIPLE  SERIES.  [§262. 


3.    Given  a  and  s  in  a  decreasing  infinite  series,  to  find 

m-  .  s — a  „        ,      a 

Ans.  m  = ,  i.  e.  1 . 

s  s 


4.    Given  a  =  1,  5  =  3,  and  n  =  oo,  to  find 


m. 


Ans.  m  =  §. 

5.  Given  m  and  s,  when  n  =  oo,  to  find  a. 

_<4ns.  a  =  (1 — /n)s, 

6.  Given  m  =  -t,  s  —  10,  and  «  =  oo,  to  find  a. 

Ans.  a  =  8. 

§  262.  (£.)  Suppose  that  at  the  end  of  one  year  from  the 
present  time,  and  also  at  the  end  of  each  succeeding  year, 
a  man  invests  a  dollars  at  r  per  cent,  compound  interest. 
What  will  be  the  whole  amount  of  his  investment  and  in- 
terest at  the  end  of  t  years  ? 

"We  shall  have  the 
amount  of  the  Jirst  investment  for  t — 1  years  =  ^(l-j-?-)'"1  „ 
"  second         "  t—2     "      =  a(l+r)'-2 ; 

•  •  •  •  •  ■ 

last  but  one    "  1  year  =  a(l-f-?")  ; 

last  "  0     "     =a. 

Hence,  if  A1  =  the  ivhole  amount,  we  shall  have 
^  =  a[(l+ry-i+(l-h-)'-2  .  .  +(HV>2+(l+'-)+l)]; 

or  (§260)  Ar=a>-^-± .  (1) 

Note.    This  is  the  amount  of  an  annuity*  of  a  dollars,  which  has 
been  forborne  (i.  e.  left  unpaid)  t  years. 

1.  Given  a  =  $100,  r  =  .06,  and  t  =  10  years  ;  to  find 
A'.  Ans.  .4' =  $1318.08. 

2.  Given  a  =  $200,  r  =  .05,  and£  =  8  years,  to  find 
A1.  Ans.  .4' =  $1909.82. 

e.)  The  present  xoorth  of  an  annuity  for  any  number  of 
years  is,  evidently,  the  same  as  the  present  worth  of  the 
amount  of  the  annuity  (§  262.  d);  i.  e.  it  is  such  a  sum,  as, 


(t)  Fr.  annuite',  yearly  payment ;  from  Lat.  annus,  a  year. 


§262.]  INTEREST. — ANNUITIES.  197 

put  at  interest  now,  will  produce  that  amount  in  the  given 

time  (§  258.  7). 

If  p>  —  the  present  worth,  we  shall  have  (§  258.  7 ;  262.  d] 

,_      A'       _a  (l+r)«-l_q/ lx  g. 

^-(1+r)«  — tr     (1+r)'         VV        1+r)'/ 

1.  Given  a  =  $100,  r=.06,  and  f  =  10  years;   to  find 
p>.  Ans.p'  =  $7S§.Q\. 

2.  Given  a  =  $500,  r  =  .06,  and  £  =  12  years ;  to  find 
.4'  and/.  -4ns.  ^£'  =  $8434.97 ;  ^'  =  $4191.92. 

/.)  If  the  annuity  be  a  perpetuity1'  (i.  e.  if  it  last  forever), 

we  have  t  =  cc,  and  (§  258.  N.  s)  t  —  0. 

.«,  j>/  =  -,  the  sum,  evidently,  whose  annual  interest  is  a. 

y.)  These  formulae,  as  well  as  those  relating  to  com- 
pound interest  (§  258.  5-9),  will  be  the  same,  whether  the 
interest  and  annuity  be  payable  at  the  end  of  each  year,  or 
of  each  half  year,  quarter,  month,  day,  hour,  or  other  peri- 
od ;  r  denoting  the  interest  of  SI  for  the  given  period,  and 
/,  the  number  of  the  periods. 

h.)  Or,  if  r=  the  interest  of  SI  for  a  year, 
t  =  the  number  of  years,  and 

n  ==  the  number  of  periods  in  a  year,  we  shall  have 

r 

-  =  the  interest  of  SI  for  the  given  period  ;*  and 

n 

nt  =  the  number  of  periods. 
.-.(§258.5)  A=p(l+?)nt.  (3) 

Also  (§  262.  d)  ]    ^  =  v[(1+D"Ll]  ;  <4> 

and  (§262..)  ^=~\}~  i}+$T]'  <«) 

Given  p  —  $100,  r  =  .06,  n  =  4,  and  t  =  3  years ;   to 
find  A.  Ans.  A  =  $119.52. 


(u)  Lat.  perpetuitas,  that  which  lasts  forever. 

*17 


198  EQUIMULTIPLE  SERIES.  [§  263,  264. 

?'.)  The  interest  may  be  conceived  to  be  payable  at  each 
moment  as  the  use  of  the  money  is  enjoyed.  In  that  case, 
n  becomes  infinite,  and  formula  (1)  reduces  to  a  peculiar 
form,  which  will  be  considered  hereafter. 

j.)  If,  while  the  interest  is  payable  annually,  the  com- 
pound interest  for  a  part  of  a  year  be  required,  the  value 
of  t  in  the  formula  of  §  258.  5  becomes  fractional. 

Thus,  the  compound  annual  interest  for  half  a  year  is 
i  I 

^(1-fr)-  ;  for  one  third  of  a  year,  p(\-\-r)3  ;  &c. 

So,  for  two  and  a  half  years,  we  have  A=zp(l-\-r)"2. 

§  263.  k.)  This  last  result  corresponds  to  the  case  in 
which  n  becomes  fractional  in  the  formula,  Z=amn-1  of 
\  258.  Nothing  prevents  our  assigning  a  fractional  value 
to  n  either  in  the  equidifferent  or  equimultiple  series. 

Thus,  in  the  series  1,  3,  5,  7,  &c,  if  n  =z  3-|-,  we  have 
/(—  a+(n—l)D=  1+2^X2  =  6. 

So,  in  the  series  1,  2,  4,  8,  &c,  if  n  =  3^,  we  have 

l(  —  am"-1)  =  1X22*=  2^=  32*=  5.65685&C      §  258. 

Note.  This,  it  will  be  observed,  is  equivalent  to  interpolating  a 
single  mean,  equidifferent  or  equimultiple,  as  the  case  may  be,  be- 
tween the  third  and  fourth  terms  of  the  series  (§§255;  265). 

/.)  Again,  n  may,  obviously,  become  zero,  or  negative. 
Thus  (§  250),  let  a  =  1,  D  —  2,  and  n  =  0. 
Then      7[  =  a+(n— 1)2)]  =  l-f-(0— 1)2  =— 1. 
If  n  =  —3,  then  I  =  l+(— 3— 1)2  =  —7. 

Also  (§  258)  let  a  =  1,  m  —  2,  and  n  =  0. 
Then         l(  =  amn~1)  =  1X20"1  =  2"1  =  |  (§  17). 
Ifw  =  — 3,  then  1=  1X2-3-1  =  2~4  =  T\. 

ci(mn 1^ 

§264.   The  formula?,    l=amn-1  (1),  s  =  -± -J   (2) 

tit         J. 

and   s  = (3)    should  be    carefully    remembered. 

m — 1 

They  contain,  it  will  be  observed,  five  quantities,  from  any 


§  265,  206.]                 INTERPOLATION.  109 

three  of  which  the  other  two  may,  obviously,  be  found 
(§  254). 

a.)  Thus,  from  the  first,  a  =         ■■  (4) 


m 
m 


=(=)*■  v 


b.)  To  find  n,  we  have  mn~x  =  -.  That  is,  n — 1  is  the 
exponent  of  the  power  to  which  m  must  be  raised,  to  pro- 
duce -.     An  equation,  in  which  the  unknown  quantity  is 

Ctl 

an  exponent,  is  called  an  exponential  equation ;  and  i^ 
solved  by  a  peculiar  process,  which  we  are  not  yet  pre- 
pared to  investigate. 

§  2G5.  c.)  From  formula  (5)  we  can  interpolate  (§  255. 
N.  r)  any  number  of  equimultiple  means  between  two  giv- 
en extremes.  For,  if  it  be  required  to  intei'polate  p  terms 
between  a  and  b,  we  shall  have  the  whole  number  of  term?, 
n,  equal  to  p-\-2.     Hence,  n— 1  =zp-\-l,  l=b, 

and  m  =  (-J  »-i  =  \-)p+1  • 

Hence  we  have  the  series, 

1.  Interpolate  2  equimultiple  means  between  3  and  81. 
Here  ^  =  2,^+1  =  3,  a  =  3,  and  J  =  81. 

m  =  (- V+ 1  =  27  J  =  3. 

Hence  the  series  is  3,  9,  27,  81. 

§266.    1.    Given  a  =  ^  wz  =  2,  and  w  =  10,  to  find  / 
and  s.  Ans.  l=z  32,  s  =  63if. 

2.  Given  a  =  l,  m  =  l,  and  rc  =  100;   to  find  /  and  s. 

Jns.  1=1,  s  =  100. 
Note.    Here  we  have 

_  a(mn—l)^ 1(1—1)  _  0 


/  _ a\nr—L)\  __  i^i— J 
A  m— 1    /         1—1 


0' 


200  HARMONIC  SERIES.  [§  267. 

apparently   indeterminate  (§  109.  c).     But  when  m  =  l,  we  have 

(§140)   " 

m» — I 

- — -  =  nmn-I  =  100x1"  =  100. 

m — 1 

3.  Given  a  =  1,  m  =  — x  (where  x  <  1),  and  n  =  ce,  to 
find  s ;  or,  in  other  words,  to  find  the  sum  of  the  decreas- 
ing infinite  series,  1 — x-\-x2 — x3-\-&c. 

Ans.  s  =■  — — • 
l-\-x 

If  we  had  x  >  1,  we  should  find  the  same  result,  but  by  a  different 

process. 

4.  Of  four  terms  of  an  equimultiple  series,  the  product 
of  the  two  least  is  8,  and  of  the  two  greatest  128.  What 
are  the  numbers?  Ans.  2,  4,  8,  16. 

5.  "What  is  the  vulgar  fraction  equivalent  to  the  repeat- 
ing decimal  .121212  &c.  ?  Ans.  -fa. 

Notes.  (1.)  This  is  the  same  thing  as  finding  the  sum  of  the  se- 
ries .12+.0012+&C.  to  infinity;  where  a  =  .12,  7/i  =  .01,  and 
n  =  co.  (2.)  In  the  same  way,  the  value  of  any  repeating  deci- 
mal may  be  found.     Thus,  we  have 

.1111  &c.  =  .1+.01+ 001+&C.  =  .l-j-(l— -1)  =  .H-.9  =  I 

6.  Find  the  vulgar  fraction  equal  to  .IOIO&c.  ;  222&C  ; 
.456456&C  :  74357435&C. 


HARMONIC  SERIES. 

i  267.  1.  Three  numbers  are  said  to  be  in  harmonic  alv 
proportion,  when  the  first  is  to  the  third,  as  the  difference 
of  the  first  and  second  is  to  the  difference  of  the  second  and 
third. 

Thus,  if  a  :  c  =  a — b  :  b — c,  then  a,  b,  and  c  are  in 
harmonical  proportion.  So  2,  3  and  6  are  in  harmonical 
proportion,  because  2:6  =  3 — 2  :  6 — 3. 

2.  Four  numbers  are  said  to  be  in  harmonical  propor- 
tion, when  the  first  is  to  the  fourth,  as  the  difference  of  the 

(»)  Gr.  dp/tovia,  joining,  harmony. 


§268-270.]        RECIPROCALS,   EQUD1FFERENT.  201 

first  and  second  is  to  the  difference  of  the  third  and  fourth. 
Thus,    6,  8,  12  and  18   are  in  harnionical  proportion ; 
because 

6  :  18  =  8-6:  18—12. 

£  268.    Let  a,  b,  and  c  be  in  harmonical  proportion. 
Then  a  :  c  =  a — b  :  b — c. 

ab — ac  =  ac — be;  (1)  or  (a-\-c)b  =  2ac. 

b  —  — —  ;    (2)  and  c  = -.     (3) 

§  269.  A  harmonic  series  or  progression  is  one  in  which 
any  three  consecutive  terms  are  in  harmonic  proportion. 

Thus,  6,  3,  2,  1.5,  1.2,  1  form  a  harmonic  series,  as  will 
be  readily  seen  by  forming  proportions  as  in  §  267.  1. 

§270.  Let  «,  b,  c,f,  g,  h,  &c,  be  consecutive  terms  of  a 
harmonic  series.     Then  (§  268.  2) 

a+P  6-h/  c+ff 

Dividing  unity  by  both  sides  of  each  equation, 

l_a+o       l_l+f      l_c+ff 

b  ~  2ac4  '    c~   2bf  f~2cg  '     C* 

o;i=I(I+Iy  ---(-+-V  i-ifi+Iv&c 

b      2\c^J'    c~2\f^b)'   f~2\g^  c)  ' 

1      1      1      1      1    „ 
•'•  -,     r,     -,     -;,     -,  &c.  are  terms  of  an  equidiflerent  se- 
tt     o      c      j      g  x 

ries  (§  252.  d).     That  is, 

The  reciprocals  of  the  terms  of  a  harmonic  series  consti- 
tute an  equidifferent  series. 

a.)  This  principle  may  be  shown  otherwise.     Thus, 

ab — ac  =  ac — be.  §268.  1. 

1111 

Dividing  by  abc, =T .  §  249. 

c       b       b       a 

b.)  Conversely,  it  can  be  readily  shown,  that  the  recip- 
rocals of  the  terms  of  an  equidifferent  series  constitute  a 
harmonical  series. 


202  HARMONIC  SERIES.  [§271. 

c)  In  order,  therefore,  to  interpolate  any  number  of  har- 
monic means  between  two  quantities,  or  to  continue  a  har- 
monic series,  of  which  two  terms  are  given,  we  have  only 
to  interpolate  a  like  number  of  equidifferent  means  between 
the  reciprocals  of  the  given  terms ;  or  to  extend  the  equi- 
different series,  of  which  those  reciprocals  form  a  part ; 
and  take  the  reciprocals  of  the  terms  so  found. 

Thus,  to  insert  two  harmonical  means  between  60  and'15> 
Ave  must  insert  two  equidifferent  means  between  ^  and 
,' .,.  This  will  give  the  equidifferent  series  ^  ^,  -b%,  £$. 
I  fence,  the  harmonic  series  is  GO,  30,  20,  1-5. 

The  succeeding  terms  of  the  equidifferent  serieswill  be 

n      fi_  _7  . 

7  0  >    <T  O  5    0  0  » 

and  the  corresponding  terms  of  the  harmonic  series, 

12,  10,  8f,  &c. 

§271.  One  of  the  most  interesting  examples  of  harmonic 
series  consists  of  the  reciprocals  of  the  natural  numbers,  1, 
2,  3,  4,5,  6,  &c.; 

viz-  l,hhhkh&*s.i 

or,  reducing  the  first  six  terms  to  a  common  denominator' 
and  taking  the  numerators, 

60,  30,  20,  15,  12,  10. 

Note.  This  series  may  be  regarded  as  the  origin  cf  the  term 
harmonical  or  musical  proportion;  the  name  having  been  applied  to 
this  series  on  account  of  the  perfect  harmony  produced  by  six  musi- 
cal strings  of  equal  thickness  and  tension,  and  having  their  lengths  in 
the  ratio  of  these  numbers.  For  the  sharpness  of  the  sound  produced 
by  a  string,  is  found  to  be  directly  as  the  number  of  its  vibrations  in 
a  given  time;  and  the  number  of  vibrations  is  inversely  as  the  length 
of  the  string.  Hence,  the  longest  string  sounding  the  key  note,  the 
second  string  will  sound  the  octave;  the  third  will  sound  the  twelfth, 
or  fifth  of  the  octave;  the  fourth,  the  fifteenth  or  double  octave;  the 
fifth,  the  seventeenth  or  third  of  the  double  octave;  the  sixth,  the 
nineteenth  or  fifth  of  the  double  octave. 


CHAPTER  X. 


PERMUTATIONS  AND  COMBINATIONS. 


§  272.  Changes  in  the  order  of  things  arranged  to- 
gether are  called  permutations'".  To  determine  the 
possible  number  of  changes  of  this  kind,  is  the  ob- 
ject of  the  theory  of  permutations. 

§  273.  A  single  individual,  as  the  letter  «,  can  obviously 
give  rise  to  no  question  of  the  kind.  But,  if  a  second  let- 
ter, b,  be  taken,  this  can  be  placed  either  before  or  after  the 
first;  thus  ab  or  ha.     Thu3, 

the  permutations  of  two  letters  =  1.2  =  2. 
Let  there  be  a  third  letter,  c.     This  may  have  three  pla- 
ces in  each  of  the  permutations  of  the  two  letters  ;  thus, 
cab,  acb,  and  abc ;  cba,  bca,  and  bac. 
That  is,  it  may  stand  before  each  of  the  other  letters, 
and  after  them  both.     Hence,  the  number  of  the  permuta- 
tions of  three  things  will  be 

2X3  (or,  for  symmetry)  1.2.8  =  6. 

In  like  manner,  a  fourth  letter  might  stand  in  four  pla- 
ces, in  each  of  the  six  preceding  permutations,  and  would 
give  the  number  of 

permutations  of  four  things  =  1.2.3.4  =  24. 
So  &  fifth  letter  might  stand  in  five  places  in  each  of  the 
(«.')  Lat.  permuto,  to  change. 


204  PERMUTATIONS.  [jj  274. 

24  permutations  of  four  letters  ;   giving  five  permutations 
for  each  change  of  the  four.     Hence,  the  number  of 
permutations  of  jive  things  =  1.2.3.4.5  =  120. 

Thus,  when  the  nth  letter  is  introduced,  there  being  n — 1 
letters  in  each  of  the  preceding  permutations,  the  new  let- 
ter can  stand  in  n  places  in  each  of  those  permutations ; 
and  we  shall  have  the  whole  number  of 

permutations  of  n  things  =  1.2.3.4.5  .  .  (?i — V)n. 

1.  How  many  permutations  can  be  made  with  the  six 
vowels  a,  e,  i,  o,  u,  and  y  ?  Ans.  720. 

2.  How  many  permutations  can  be  made  in  writing  the 
nine  digits?  Ans.  362880. 

Note.  The  expression  [«]  is  sometimes  used  to  denote  the  pro- 
duct 1 .  2  .  3     .     .     n.     Thus   [10]=1.2.3.4.5.6.7.S.9.10. 

§  274.  We  sometimes  inquire  the  number  of 
changes  in  the  position  of  n  things  taken,  not  all  at 
once,  but  a  part  at  a  time.  The  changes  thus  pro- 
duced are  called  arrangements  or  variations. 

To  find  the  number  of  such  arrangements,  we  must  con- 
sider that  we  may  write  any  one  of  the  n  letters,  as  a,  be- 
fore each  of  the  remaining  n — 1  letters.  We  shall  thus 
have  n — 1  arrangements  of  n  letters  taken  two  and  two,  ia 
each  of  which  a  stands  first.  "We  may,  in  like  manner, 
have  n — 1  arrangements  in  which  b  shall  stand  first ;  and 
so  for  each  of  the  n  letters.  Hence,  the  whole  number  of 
arrangements  of  n  things  taken  two  at  a  time  will  be 

n(n — 1). 

Again,  each  of  these  n(?i — 1)  arrangements  of  n  letters 
taken  two  at  a  time  can  be  placed  before  each  of  the  re- 
maining n — 2  letters.  Hence,  the  number  of  arrangements 
of  n  things  taken  by  threes  will  be 

n(n—l)(n—2). 

In  the  same  way,  placing  each  of  these  »(n — l)(n — 2) 
arrangements  before  each  of  the  remaining  n — 3  letters,  the 


§  275.]  ARRANGEMENTS.— COMBINATIONS.  205 

number  of  arrangements  of  n  things  taken  four  at  a  time 
will  be  «(«— l)(n— 2)(n— 3). 

And,  in  general,  the  number  of  arrangements  of  n  things 
taken  p  at  a  time  will  be 

n(n — l)(n — 2)     .     .     (n—p-\-Y)  ;    or  [n,  n— £>+l], 
by  a  notation  analogous  to  that  of  §  273.  N. 

a.)   We  have,  obviously, 

That  is,  the  number  of  arrangements  of  n  individuals  ta- 
ken p  at  a  time  is  equal  to  the  whole  number  of  permuta- 
tions of  n  individuals,  divided  by  the  number  of  permuta- 
tions of  n — p  individuals  ;  (i.  e.  by  the  number  of  permuta- 
tions which  can  be  made  with  the  individuals  left  out  of 
each  arrangement). 

1.  How  many  arrangements  can  be  made  with  the  10 
Arabic  numerals,  taken  2  at  a  time  ?  Ans.  90. 

2.  How  many,  if  they  be  taken  3  at  a  time  ? 

Ans.  720. 

3.  How  many  arrangements  can  be  made  from  the  72 
numbers  of  a  lottery,  taking  3  numbers  upon  each  ticket  ?    \ 

Ans.  357840. 
b.)  I£p  —  n,  we  shall  have  simply  the  permutations  of  n 
things  =  n(n—l)     ....     2  . 1,  as  in  §  273. 

§  275.  Combinations  are  the  groups,  that  can  be 
formed  of  individuals  without  reference  to  the  order 
of  arrangement ;  in  other  words,  groups  such,  that 
no  two  of  them  shall  be  composed  of  the  same  individ- 
uals. Thus  ab  and  ba  form  two  permutations  or 
arrangements,  and  but  one  combination. 

And,  in  general,  whatever  be  the  number  of  things,  on- 
ly one  combination  can  be  formed  by  taking  them  all  at 
alg.  18 


206  PERMUTATIONS  AND  COMBINATIONS.  [§  275. 

once.  For  two  combinations  are  not  different,  unless  they 
differ  in,  at  least,  one  of  the  individuals  contained  in  them. 

Hence,  each  combination  of  n  things  may  be  subjected 
to  1 .  2  .  3  .  .  n  permutations,  without  affecting  the  combi- 
nation. So,  if  we  combine  n  things,^?  at  a  time,  each  com- 
bination admits  of  1.2.3 p  permutations  or  ar- 
rangements. 

Hence  we  shall  have  only  one  combination  for  every 

1.2.3 p      arrangements.     If  then  we  divide  the 

number  of  arrangements  by  the  number  of  permutations  in 
each  arrangement,  we  shall  have  the  number  of  combina- 
tions. 

That  is,  if  we  combine  n  things,  p  at  a  time,  we  shall 
have 

_T  ...  No.  arrangements  of  n  things  taken  »  &» 

INo.  combinations  — — . 

No.  permutations  of  p  things. 

."  •  the  number  of  arrangements  of  n  things  taken  p  at 
a  time  is  (§274) 

n(n — 1) (ii—p-\-\)  ;     or  [n,  n— £>+l]  ; 

and  the  number  of  permutations  of  p  things  is 
1.2.3  .  .  .  .  p ;   or  \_p~\. 
Therefore  the  number  of  combinations  of  n  things  taken 
p  at  a  time  is 

n(n — 1)     .     .     .  (n—p-\-l)  _   [n,  n—p-\-\~\ 

1.2.3     7~.     \     .~~ p  "  [p] 

a.)  If  the  letters  denote  algebraic  quantities,  the  number 
of  combinations  of  n  letters  taken  p  at  a  time  is,  evidently, 
the  same  as  the  number  of  distinct  products  of  the  quanti- 
ties taken  p  at  a  time. 

b.)  If  n  things  be  taken  p  at  a  time,  then  (§$  274.  a ;  275) 
n(n— 1 ) .  .  (n— H-1 )  (n—p)  ..3.2.1 


No.  of  arrangements  = 
No.  of  combinations  = 


1.2.3     .     .     {n—p) 

»(n — 1)  .  .  (n— p4-l)(n— p)  .  .3.2.1 

1.2.3...^.    1.2.3...  {n—p)  ' 


§  275.]  COMBINATIONS. — PRODUCTS.  207 

Again,  if  n  things  be  taken  n—p  at  a  time,  we  have 

72(71—1)  .  .    (Jl— p-\-l)  («'— p)  .  .  3  .  2  .  1 


No.  of  arrangements  =: 
No.  of  combinations 


1.2.3.  .  .  .  p 

n(»— 1 )  .  .  {n—p-\-l )  (n—p)  ..3.2.1 


1.2.3  .  .  .  p .1 .2.3  .  .  .  (n — p) 

Hence,  the  number  of  combinations  of  n  individuals,  is 
the  same  whether  they  be  taken  p,  or  n—p,  at  a  time. 

Thus,  the  number  of  combinations  of  10  letters  will  be 
the  same,  whether  they  be  taken  3  and  3,  or  7  and  7. 

c.)  The  last  principle  is  evident  also  from  the  fact,  that, 
for  each  combination  of  p  things  taken,  a  combination  of 
n—p  things  must  be  left. 

1.  How  many  products  (§  275.  a)  can  be  formed  of  the  6 
quantities,  a15  a2,  a3,  «4,  a5,  a6,x  by  taking  them  1  by  1, 
2  by  2,  3  by  3,  4  by  4,  and  5  by  5  ? 

Ans.  Q,  15,  20,  15,  and  6. 

2.  How  many  products  of  4  quantities  taken  1,  2,  3,  and 
4,  respectively,  at  a  time?  Ans.  4,  6,  4,  and  1. 

d.)    The  number  of  combinations  must,  of  course,  be 

a  whole  number.      Therefore       '       •%  is  a   whole 

number. 

(x)  These  numbers,  1}  2,  3,  &c,  are  used  as  accents  (§  1.  e). 


CHAPTER  XL 


UNDETEKMINED  COEFFICIENTS. 


§  276.  Let  the  equation 

Aa?-\-BaP+  <7af-f-&c.  =  A>xi»-\-B>x*'+  CV+&C.  (1) 
be  true  for  all  values  of  x  (i.  e.  whatever  value  may  be  as- 
signed to  x) ;  A,  B,  G,  &c.,  and  A1,  B',  C,  &c,  being  any 
quantities  whatever  not  equal  to  zero  or  infinity,  and  each 
member  of  the  equation  being  arranged  according  to  the 
ascending  powers  of  x. 

It  is  required  to  determine  the  relation  existing  between 
the  exponents,  and  also  between  the  coefficients  of  x  in  the 
corresponding  terms,  on  the  two  sides  of  the  equation. 

Dividing  both  members  of  the  equation  by  xp,  we  have 

A+Bx*-*-t-  Cxr-P+&c.  =  A'xp'-p-\-Bxv-p+  G'xr'-*'+&c. 

Now,  as  this  equation  is  true  for  all  values  of  x,  it  i3  true 
when  x  =  0.  But  if  x  =  0,  the  first  member  reduces  to  A 
(the  exponents  of  x  in  all  the  terms,  but  the  first  being  >  0, 
i.  e.  positive) ;  and  the  second  member  evidently,  becomes 
zero,  if  jt/> p  ;  infinite,  it'  p'<p. 

Hence,  if  p'~>p,  we  shall  have 

A  —  0,  which  is  contrary  to  the  hypothesis  : 
and,  if  p'<p,   A  =  co,  also  contrary  to  the  hypothesis. 

We  must,  therefore,  have p'=.p ;  which  gives^/ — p  =  0 ; 

and,  if  x  =  0, 

A  =  A'. 


§277-280.]     UNDETERMINED  COEFFICIENTS.  209 

Hence,  removing  equal  quantities  from  both  sides  of  (1), 
Bx*+Cxr+&c.  =  B'x«+Cfxr,-{-&c.  (2) 

Dividing  by  xq,  and  making  x  =  0,  we  shall  prove  q  =  q', 
zn<lB=B'.  And,  in  like  manner,  r  =  r',  C—C;  &c« 
Hence, 

§  277.   If  an  equation  between   two  polynomials, 
functions  of  x,  be  true  for  all  values  of  x,  it  must  have 
like  powers  of '  x  on  both  sides;  and  the  coefficients  of 
the  like  powers  must  be  severally  equal  to  each  other. 

§  278.  a.)  Let  the  equation  be  given  of  the  form, 

A-\-Bx+Oe*+&c.  =  A,+B'x+C'x2+&c.         (3) 
Then,  making  x  =  0,  A=zA'; 

and  canceling  A  and  A'  in  (3),  dividing  by  x,  and  again 
making  x  =■  0,  B=B';   &c. 

§  279.  b.)  Or,  again,  transpose  all  the  terms  of  equation 
(3)  to  the  first  member,  and  arrange  with  reference  to  the 
powers  of  x.     Thus, 

A—A'-\-{B—B')x-\-{  G—  C>)xs-$-&c.  =  0.         (4) 
Making  x  =  0,         A— A'  =  0  ;  and  .-.  A  —  A'. 
Then  (B—B')x+(C—Cr)x*+&c.  =  0 

Dividing  by  x,         B—B>-\-{  C—  C')x+&c.  =  0. 
Making  x  =  0,     B—B'  =  0  ;  and  B  =  Bf ;  and  so  on. 

Represent  A— A'  by  31;    B—B'  by  If;    C—C  by  P; 
&c     Then  if  we  have,  for  all  values  of  x, 
M-\-Nx+Px-+&g.  =  0, 
we  shall  have  also 

M=Q;  y=0;  P=0;  &c.     Hence, 

§  280.  If  any  polynomial  of  the  form,  M+Nx+Px2 
-\-SfC,  be  equal  to  zero  for  all  values  ofx,  each  of  the 
coefficients  of  the  several  powers  of  x,  must  be  sepa- 
rately equal  to  zero. 

*18 


210  UNDETERMINED  COEFFICIENTS.  [§281. 

c.)  An  equation,  which  is  true  for  all  values  of  a  varia- 
ble, is  said  to  be  true  independently  of  the  variable.  Such 
an  equation  is  an  absolute  equation  (§  37.  d). 

Thus,  the  equation, 

(a-\-z)  2  =  a2+2ax-{-x2, 
is  true  independently  of  x.     On  the  other  hand,  the  equa- 
tion, \-\-x-  =  2x—x3, 

may  be  true ;  but  its  truth  depends  on  the  value  given  to  x 
(§38). 

§  281.  The  above  principle  (§§  276-280)  is  the  founda- 
tion of  the  method  of  undetermined  coefficients  ;  a 
method  of  very  great  utility  in  the  development  of  func- 
tions and  the  investigation  of  principles. 

1.    Develop  — - —  into  a  series. 
l-\-x 

Assume       ■——=Ax~1-}-Bx0+Cx+Dx2+&c, 
Then,  if  x  —  0,  we  have  1  =  oo  ;    which  is  absurd. 

Again,  assume  =  Ax-{-Bx2-\-Cx3-\-&c. 

Then,  if  x  =  0,  we  have  1  =  0;  which  is  absurd. 

Assume  then  =  A-\-Bx-\- Cx2-\-Dx3-\-&c. 

Clearing  of  fractions  and  transposing. 


o> 


0  =  A    +A    x+B    x2+C    a;3+&c. 
—1     +5      +<7        +Z> 

.-.    .4—1=0;   A+B=0;  B+C=0;    (7+2)=  0;  &c. 
A  =  l;      B——A  —  —\;      C=—B=l; 
Z>  =  —  C=—  1;    &c. 
Introducing  the  values  of  A,  B,  C,  &c,  we  have 

—  1— x+x2—  a;3+x4—  z5+&c. 

1+a:  ~  n 

Note.  The  results  of  the  several  suppositions,  in  this  instance, 
indicate  the  method  of  ascertaining  the  form  of  the  series  to  be  as- 
sumed.    We  may  generally  determine  this,  before  writing  the  seiies, 


§281.] 


SKR1ES. PARTIAL  FRACTIONS. 


211 


by  making  x=:0  in  the  function  to  be  developed.  The  series  must 
be  taken,  so  as  to  become  finite,  infinite  or  zero  for  x-  =  0,  according 
as  the  function  becomes  finite,  infinite  or  zero  for  the  same  value  of  x. 

2.    Develop  (a—x)-1.     See  §87.  c.  2. 

Assume       -±-  =  A-{-Bx+Cx2-\-Dx3+&c. 
a — x 

Then(§4G)    l=Aa—A 

-\-aB 


x—B 

-\-aC 


xn—C 
-\-aD 


X3 — &c. 


Aa  —  1;    aB—A  —  0;    aC—B  =  Q;    aD—G=0;   &c. 


A  =  1-,   B  =  ±  =  L.,    C=B-  =  K;   B: 


a 


a       a- 


a       (I'- 


ll 


-;  «fec. 


...  _J_=i  +  la+4«s+&c.  =  -(l+-  +  ^-h&c.) 
a—x      a      a-        a3  a\        a      a-  / 

Or      (a— x)-1  =«-1+a-2«+a-3x2+«-ix'3+&c. 

=  a-i(l+a-ia:+a-2a;2+&c.) 

i 
3.    Develop  (a — x)2. 

Assume       (a—x)  *  =  A-{-Bx-\-  Cx2-\-Dx3-{-&c. 


Squaring,     a—x  =  A2+2ABx+B2     x2+2BC 

■\-2AG         +2AD 
A2  —  a;  2AB  =  —  1;  B2+2AC=0; 
2BCJr2AD  =  0;   &c. 


a;3+&c 


i                    11 
A  =  a2;    B  =  --= -: 

Z  2a- 


(a — xy2  —  a"  — 


1   x 


lx2 


C= 


±a^ 


2.4a2 


;    &c. 


1     x- 


2  a2      2.4a2      2.2. 
!  1       x* 


&c. 


1  X 

2~a~Y74~a2      2.2 
3a;— 5 


^-&c.Y 
a*        y 


4.   Decompose 


into  fractions,  whose  sum  is 


x2—Gx+8 

the  given  fraction,  and  whose  denominators  are  the  factors 
of  the  given  denominator. 

a;2_6a;_|_8  —  (x—4)(x— 2).  §  213.  1. 


212  UNDETERMINED  COEFFICIENTS.  [$  281. 

Therefore  assume 

3x— 5  A      .     B 


x2— 6x+8      x— 4  ^  a;— 2 ' 
3x-5       _^(,;-2)+i?(a:-4) 

.-.  3a:— 5  —  A{x— 2)-\-B(x— 4)  =  (J+-B)*—  (2^4+45). 
.-.     ^+5  =  3,   2A+iB  =  5.     .-.   ^=|,   B-  —  \. 
3a:— 5  7  1 

X2_qx+s~2(x^      2^-2)'     ^ee§118.  3. 

Otherwise ;  as  the  equation, 

3a:— 5  =  A[x— 2)-\-B(x— 4), 

is  true  for  a#  values  of  a:,  it  is  true,  when  x  =  2  (i.  e.  when 
a:— 2  =  0). 

Introducing'this  value  of  x,  we  have 

6—5  =  5(2—4)  ;  and  .-.  B  =  —  £. 
Again,  if  a:  =  4  (or  a: — 4  =  0),  we  have 

12— 5  =  J(4— 2)  ;  and  .-.  A  =  l,aa  before. 

j.     _  a34-bx2  /  a3-\-bx2      \ 

o.   Decompose  —^ -(  =—. -w    ,     ). 

a2x — x3  v      x{a — x)(ci-\-x/ 

.       a  ,      a4-b  a-\-b 

Am--*  +  Z^)-2^)-    See  §118.  2. 

6.^yelop(l^)-(=^)-=iTA_)    in    an 

infinite  series. 

Ans.  1— 2a:+3x2— 4a:3-f&c.     Compare  §  87.  c.  5. 

3x2— 1 


7.   Decompose 


a:(a;+l)(x— 1) ' 

Ana. 


x      a:-|-l       a: — 1 

l—x 
3.    Develop  — —  in  an  infinite  series. 
1-j-x 

Ans.  1— 2a;+2a;2— 2a;3+&c. 


CHAPTER  XII. 


BINOMIAL  THEOREM. 


§  282.  Any  positive  integral  power  of  x-\-a  can  be  found 
by  multiplying  x-\-a  into  itself  the  requisite  number  of 
times  (§  164).  The  proper  combination  of  this  process 
with  division  and  with  the  extraction  of  roots  will  give 
negative  and  fractional  powers  (§  163). 

But  this  process,  when  applied  even  to  positive  integral 
powers,  beyond  a  few  of  the  lowest,  becomes  tedious ;  its 
application  to  negative  and  fractional  powers  would  be  ex- 
tremely inconvenient. 

The  Binomial  Theorem  enables  us  to  find  im- 
mediately any  power  of  a  binomial,  whether  the  ex- 
ponent be  positive  or  negative,  integral  or  fractional. 

§  283.  Let  it  be  required  to  find  the  nth.  power  of  x-\-a. 

I.    Let  re  be  a  positive  integer. 

Then  the  nth.  power  of  x-\-a  is  the  product  of  n  factors 
each  equal  to  x-\-a ;  i.  e. 

(x-\-a)n  =  (x-\-a)(x-\-a)(x-\-a)  .  .  .  to  n  factors. 

To  find  how  the  terms  of  these  factors  are  combined  in 
the  terms  of  the  product,  multiply  together  n  unequal  fac- 
tors, x-\-ax,  x-\-a2,  x-f-«3j #+«»• 

Then  (x-\-a1)(x-\-a2)  =  x-Jra1  x-\-axa.2. 

-\-a2 

{x-\-al){x-\-a2){x-\-a^)^=x^-\-al  x2-\-ala2  x-\-a^a2ay 

-\-a2      -\-a1a3 
-\-a3      -j-«2«3 


214 


BINOMIAL  THEOREM. 


K  28^ 


(x+^i)(^+a2)(x+a:i)(x-\-aJ  = 


+  «4 


+  «1«3«4 
-}-«2a3<r'f4 


a:-)~aiff203a4 


X3-|-«1«o 
+  «1«3 
+  «1«4 
~ha2f/3 
+  «2°4 
+  «3«4 

Hence  ire  find,  that,  so  far  as  we  have  proceeded,  (1.) 
The  exponent  of  a:  in  the  first  term  is  equal  to  the  member 
of  factors  ;  and  (2.)  diminishes  by  unity  in  each  of  the  fol- 
lowing terms  till  it  becomes  zero;  also  (3.)  the  coefficient 
of  x  in  the  first  term  is  unity ;  (4.)  in  the  second  term  it 
is  the  sum  of  the  second  terms  of  the  binomial  factors ;  (5.) 
in  the  third  term,  the  sum  of  the  products  of  the  second 
terms  taken  two  and  two  ;  (6.)  in  the  fourth  term,  the  sum 
of  the  products  taken  three  and  three  ;  and  so  on,  that  in 
the  last  term  being  the  product  of  all  the  second  terms. 

To  show  the  universality  of  this  law,  let  us  suppose  that 
we  have  found  it  true  for  n — 1  factors,  and  see  whether  it 
will  hold  good  for  n  factors  (§  9G.  N.  1).     Thus,  suppose 
(x-\-al)(x-\-a2)(x-\-a3)  ....   (ar-L-a^)  = 


+  «»-! 


xn~'2-\-ala2 

+  «1«3 


xn~3-\-ala2a3\xll—i..-\-a1a2..an_] 
-\-axa2aA 
&c. 


T"  an-2Un-l 

Now  introducing  the  nth  factor, 
(x+ax)(x+a2)(x+a3)   .  .   (x+an)  = 


xn-\-al 

-J-«2 


xn~1-^-a1a2 


xn~2-\-a1aia3 

+  «i«2«4 

&c. 


xn-3..-f-0!1a2. 


.a, 


»« 


(1) 


§284,285.]     positive  integral  exponent.  215 

Hence  (§  283.  1-6), 

§  284.  (1.)  The  law  of  the  exponents  is  obviously  the 
same  as  before. 

(2.)  The  coefficient  of  a;  in  the  first  term  is  unity,  as  be- 
fore. 

(3.)  The  coefficient  of  x  in  the  second  term  is  the  sura 
of  the  second  terms  of  the  n  factors. 

(4.)  The  coefficient  of  x  in  the  third  terra  is  the  sura  of 
the  products  of  the  n — 1  second  terms  taken  two  and  two, 
and  also  of  the  products  of  those  n — 1  terms  by  the  new 
term  a„ ;  hence  it  is  the  sum  of  the  products  of  the  second 
terms  of  the  n  binomials  taken  two  and  two. 

(5.)  The  coefficient  of  x  in  the  fourth  term  is  composed 
of  the  several  products  of  the  n — 1  second  terms  taken 
three  and  three,  and  also  of  their  products  taken  two  and 
two  multiplied  by  the  new  quantity  an ;  i.  e.  it  is  the  sum 
of  all  the  products  of  the  n  second  terms  taken  three  and 
three. 

(G.)  The  last  term  is  evidently  the  product  of  all  the  sec- 
ond terms  taken  together ;  or,  which  is  the  same  thing,  the 
sum  of  the  products  of  the  n  second  terms  taken  n  and  n. 
For  there  can  be  only  one  such  product  (§  275). 

Hence,  if  the  above  law  is  true  for  n — 1  factors,  it  is  true 
for  n  factors.  But  we  have  seen,  that  it  is  true  for  4  fac- 
tors ;  it  is  therefore  true  for  5,  for  6,  7,  &c.  That  is,  it  is 
universal. 

§  285.  Now,  if  av  a2,  a3,  .  .  .  .  an  are  each  equal  to  «, 
the  coefficient  of  x  in  the  second  term,  will  be  na ;  each 
term  in  the  third  coefficient  of  x  will  be  a2  ;  each  term  in 
the  fourth,  a3  ;  and  so  on;  each  term  in  the  nth.  coefficient 
of  x  being  a"-1. 

Moreover,  a2,  in  the  coefficient  of  x  in  the  third  term, 

will  be  repeated  as  many  time3  as  there  can  be  products  of 

n(n — 1) 
n  quantities  taken  two  and  two  :  that  is  (§  275.  a). 

JL  .  a 


216  BINOMIAL  THEOREM.  [§  285. 

Also  «3,  in  the  coefficient  of  x  in  the  fourth  term,  will 

be  repeated  as  many  times,  as  there  can  be  products  of 

,         ,  ,    ,  ,  n(n — l)(?i — 2) 

n  quantities  taken  three  and  three  ;  that  is  — ~ -. 

J.  •  Z  .  o 

Hence,  we  shall  have  (n  being  a,  positive  integer) 

71(71 — 1^ 
(x+ayi'  =  xn-\-nxn-ia-\-     ,    0  yx"-sa2-l-&c +0". 

That  is,  (1.)  the  first  teim  of  any  positive  integral  power 
of  a  binomial  is  equal  to  that  power  of  the  first  or  leading 
term  of  the  binomial;  (2.)  the  exponent  of  the  first  term  of 
the  binomial  diminishes  by  unity,  till  it  becomes  0  ;  and  the 
exponent  of  the  other  term  increases  by  unity  from  0  to  n. 

(3.)  Ths  coefficient  of  the  first  term  is  unity ;  and  (4.) 
that  of  the  second  term  (i.  e.  of  both  x  and  a)  is  n. 

(5.)  The  coefficient  of  any  term  whatever  after  the  first  is 
found  by  multiplying  the  coefficient  of  the  preceding  term  by 
the  exponent  of  the  leading  quantity  in  that  term,  and  divid- 
ing by  the  number  of  terms  preceding  the  required  term. 

a.)  The  exponent  of  the  leading  quantity  becoming  0  in 
the  (?i-(-l)th  term,  the  next  coefficient  found  will  be  0  ;  and 
the  series  will  terminate,  consisting,  as  is  evident,  of  n-\-l 
terms. 

b.)  The  number  of  combinations  of  n  things  is  the  same, 
whether  they  be  taken  p  and  p,  or  n — p  and  n — p,  (§  275.  b). 

Hence  the  coefficient  of  the  term,  which  has  p  terms  be- 
fore it,  is  equal  to  the  coefficient  of  the  term,  which  has 
n — p  terms  before  it,  or  p  terms  after  it. 

Consequently,  if  we  find  the  coefficients  of  the  first  half 
of  the  terms,  we  have  also  the  coefficients  of  the  last  half 
in  the  reverse  order. 

c.)  The  last  remark  is  also  evidently  true,  from  the  fact, 
that  (a-\-x)n=  (x-\-a)n,  and  there  is  no  reason  why  we 
should  begin  with  x11  rather  than  with  an.     Thus, 

(a+g)"  =  a"+waB-1a;-f  W^~^a"-gx8+     .     .     .      +x\ 

1    »    — 


§286,287.]      POSITIVE  INTEGRAL  EXPONENT.  217 

§  286.    1.   What  is  the  square  of  x-\-a? 
Here  we  have  n  =  2. 
»•.      (x-\-a)*  =  (x+a)  2  =  xn-\-nxn-ia+n\^xn-*a* 

JL  •  ■*» 

+  K«-l)(^-2)xn_3a3+&c.  =  s.+toH-  Mzofl, 

—  x*-\-2xa-\-a*. 

2.  (a;-|-a)6  =  what? 

Here  «  =  6. 

6.5  6.5.4 

(x-]-a)6  zr:a;6-{-6a:5a-(---^--a:4a2-}-     '     '  nx3as-\- 

1  .  &  J.  •  ^  .  o 

6.5.4.3  g  A  ,  6.5.4.3.2     s  ,  6.5.4.3.2.1 

1.2.3.4*  a  +1.2.3.4.5Xa  +  1 .  2  .  3  .  4.  5  .  6® 

=  .r"  |  i;x5«-}-15a;4a2-f20x3a3-}-15a;2a4+6a:a5+ae. 

3.  i>+a:)3  =  what?     (a:-(-a)4?     (1+ar)*? 

§  287.   d.)  If,  in  the  general  formula  (§  285.  c),  we  put 
— x  for  -\-x,  we  have 

(«— x)n=  an— nan~\x-\-     \.~  *  a*~*x\— &c. ; 

the  terms  containing  the  oc?c?  powers  of  aj  being  negative, 

1.  (a— a;)5  :=  what?     (a— x)2?     (a— x)3? 

2.  (a2— x2)5  =  what? 

Jws.  (a2)s— 5(a2)*x2+10(a2)3(x2)2— 10(a2)2(a:2)3+ 
5(a2)(x2)4 — (a;2)5;  or,  reduced, 
ai  °— 5«8a;2H-10aGa;4— 10a4a;6-l-5a2a;8— a;1 ». 

3.  (a;2±2oa?)3=what? 

Jns.  a;e±6aa;5-j-12a2a;*±8a3a:3. 

Make  x2  =  6,  and  2ax  =  c;  develop  (6+c)3,  and  substitute  the 
values  of  6  and  c. 

e.)  The  formula?  (§  285.  c ;  287.  rf)  may  be  put  under 
another  form.     For 

a±x=za(l±-Y     .-.  (a±x)n=an(l±~y, 

ALG.  19 


218  EIX03IIAL  THEOREM.  [§288,289 

,   j_  \n        nft^l  x  i   n(n — 1)  x2   .  n(n—l)(?i—2)  X~' 

.'.    (a±x)n  =  a"[  l±n-4-  — — — y  —  ±  -^ y-i L  — 

;  V         aT      1.2     a2  1.2.3        a-: 

w(??— l)(n— 2)  .  .  (»— 2jp-fl)  «2^  n 

H  1.2.3  .  .:(2p~l)2p         ^  / 

Note.  The  above  demonstration  proceeds  upon  the  supposition 
that  n  is  a  positive  integer ;  and  is,  of  course,  applicable  to  that  case 
only. 

§  288.  II.  Whatever  be  the  value  of  n,  whether  integral 
or  fractional,  positive,  or  negative,  let  it  be  assumed,  that 

(x+y)n  =  A+By+  Cy?+Dy*+JEy*+&c. ;         (1 ) 

.4,  B,  G,  &c,  being  functions  (§  26)  of  aj  and  w,  and  entire- 
ly independent  of;?/. 

Note.  There  can  be  no  negative  powers  of  y,  because  (x+y)n  is 
not  necessarily  infinite  when  y  =  0.  There  must,  moreover,  be  a 
term  containing  y®,  because  (x+y)n  is  not  necessarily  zero,  when 
y  =  0  (§281.  N.). 

§  289.  1.  Let  n  be  a  positive  fraction,  -  (p  and  q  being 
both  integers). 

Welrave  (x-\-y)*  =  w '(l+  -V '.  §  287.  e. 

Assume      (l-f-^V  =  l+P'-+&c,       for  all  values  of -. 
\        x/  a;  a; 

Then  (l+^)P=(l+p|+&c.)*.  §  52.  N. 

.-.  (§  285)  l+;>^  +&c.  =  1+?P-  +&c. ; 

the  remaining  terms  on  both  sides  containing  only  higher 

c  y 

powers  or  -. 

JO 

.-.(§277)  p  —  qP',    or  P=?-  =  n. 

(x+y)«  =  xi  (1+*-  -  +&c.)  =  xi  +t-xz     y+&c 
v       *'  q  x  q 

Hence,  in  the /rsi  ftw  terms,  the  same  law  prevails  with 


§  290,  291.]    NEGATIVE  EXPONENT. — INCREMENT.  219 

the  positive  fractional,  as  with  the  positive  integral  expo- 
nent {§  285.  1,  3,  4). 

2.    Let  n  be  negative.     Then 

(^i-y)     —  (x_|^)n  —  xn-{-nx"-1y-\-&c.' 

(x-\-y)-n  =  x~n — ?ix~"-1y-\-&c,  by  division  ; 

tJie  remaining  terms,  evidently,  containing  successively  low- 
er powers  of  a;  and  higher  powers  of  y. 

Hence,  again,  the  first  two  terms  follow  the  same  law 
with  the  negative,  as  with  the  positive  exponent. 

Hence,  universally,  whatever  be  the  value  of  n  (i.  e. 
whether  it  be  positive  or  negative,  integral  or  fractional), 

;0.    The  first  two  terms  of  (x-\-y)n  are  xn-\-nxn~1y. 

We  have,  therefore,  in  the  series  (1)  of  §  288, 

A  =.  xn,  and  B  =  ?ixn~1  ;  and  the  series  may  be  written 

(x-{-y)n  =  a;"-f?2x"-iy+  CyS-\-Dy*-\-Ey*-\-&c. 

§  291.  Let  now  each  of  the  quantities  x  and  y  be  suc- 
cessively increased  by  any  quantity  whatever  k.  The 
function  (x-\-y)n  will,  obviously,  undergo  an  equal  change 
in  each  case ;  i.  e. 

Note.    The  quantity,  h,  by  which  x  and  y  are  increased,  is  cal- 
led an  increment^  of  a;  and  y. 

1.  Adding  h  to  y,  series  (1)  of  §  288  becomes 

[»+(*+*)]"  =  A+B(y+h)+  C(y+h)  *-\-D(y+h)3-h&C. ; 

or  ix+(y+h)Y  =  A+By+Cy*±By*+By±+&C.         (2) 

-\-Bh-\-2  Cfyk+8Ih/2h-l-4By3h-\-&c. ; 
&c.       &c.        &c. 

writing  only  the  terms  containing  A0  and  h1. 

2.  The  substitution  of  x+h  for  x  will,  of  course,  produce 
no  change  iny,  or  in  the  manner  in  which  it  enters  into  the 

ly)  Lat.  inerementum,  an  increase. 


220  BINOMIAL  THEOREM.  [§  292. 

expression ;  but  it  will  produce  a  change  in  each  of  the  co- 
efficients, A,  B,  O,  &c.  For,  as  these  coefficients  are  func- 
tions of  x,  they  will,  in  general,  change  their  value  when- 
ever the  value  of  x  is  changed.  Therefore,  the  powers  of 
y  remaining  as  they  are,  their  coefficients  will  be  what  A, 
B,  C,  &c.  become,  when  x  is  changed  into  x+h ;  i.  e.  they 
will  be  the  3ame  functions  of  x+h,  as  A,  B,  C,  &c.  are  of  x. 
Representing,  then,  by  Ax+ h,  B^,  C^h,  &c,  the  values 
assumed  by  A,  B,  C,  &c,  when  x  becomes  x-\-h,  we  shall 
have 

[(»+*)+*?  =  A^+B^jrt- Cx^?,"-+nx+fty3+&c.  (3) 

$  292.   Now  we  have  already  found 

A  =  x",  and  B  =.  nxn~1. 
Ax+h  =  (x+hy ;  and  Bx+h  =  n(x+h)n-K  (4) 

But  we  do  not  know,  what  functions  C^,  JDx+h,  &c, 
are  of  x-\~h,  because  we  do  not  know  what  functions  G,  Df 
&c.  are  of  x.  In  other  words,  we  do  not  know  what  C,  D, 
&c.  will  be,  when  x-\-h  is  substituted  in  them  for  x,  because 
we  do  not  know  what  they  are  now. 

Let  it  be  assumed,  then,  that 

Cx+h  =  C+  C'h+&c. ;  Dx+h  =  D+iyh+&c.  j        (5) 
and  so  on ;  and  assume,  for  symmetry 

Ax+h[=z  (*+*)•]  =  A+A'h+&c. ; 
and  Bx+h[=  n  (x-\-h)n~  *  ]  =  B+B  'h+&c. 

Notes.  (1.)  This  supposition,  evidently,  involves  no  absurdity 
(§281.  N);  for,  when  A  =  0,  the  expressions  (5)  severally  reduce 
to  C,  D,  &c,  as  they  ought,  being  then  simply  functions  of  x  as  at 
first  (§288). 

(2.)  It  will  be  observed,  that  A,  B,  C,  &c,  in  these  assumed 
values,  are  the  primitive  undetermined  coefficients,  functions  of  x 
(§288);  and  that  A',  B>,  C,  &c  are  the  coefficients  of  h*  in  the 
several  expressions,  when  x+h  is  substituted  for  x. 

(3.)  If  the  variable  of  a  function  is  increased,  and  the  function 
developed,  the  coefficient  of  the  first  power  of  the  increment  is  a 
quantity  very  much  employed  in  analytical  investigations;  and  is  cal- 
led the  first  derived  function,  or  the  first  derivative,  or  derivate,  or 


'  293.]  DERIVED  FUNCTIONS. 

differential  coefficient,  of  the  primitive  function.     Thus  A1  is  the 
first  derived  function,  or  derivate  of  A;  B',  of  B;  &c. 

(4.)  In  like  manner,  if  x+/i  be  substituted  for  x  in  A',  B' ,  &c, 
the  coefficient  of  At  is  the  first  derived  function,  or  derivate  of  A',B', 
&c;  and  may  be  called  the  second  derived  function,  or  second  de- 
rivate of  A,  B,  C,  &c;  and  may  be  represented  by  A",  B",  &c 
The  same  process  deduces  from  the  second  derivate,  a  third,  A'", 
B'",  &c;  from  the  third,  a  fourth,  A"",  B"",  &c;  and  so  on. 

§  293.  Substituting  for  A^,  Bx+fl,  Ox+ft,  &c.,  the  val- 
ues (5)  assumed  above  (§  292),  and  writing  as  in  (2)  of  §  291 
only  the  terms  containing  h°  and  h1,  we  have 

t(x+h)+j/]H  =  A+A>h+&c.+(B-\- B'k+&c.)y-\-(  C-\-  C'h 

+&e.)y2+&c; 

or,  arranging  according  to  the  powers  of  h, 

I(»+*)+yr=^+iH-^»+^'+^*-f*c-      (6) 

+(A'+B'y+CY+D'y3+&c-)h 

-f&c. 

Equating  (§  291)  the  second  members  of  (2)  and  (6), 

A+By-\-Cy'*+&c.         )       (A-\-By+Cy*+&c.  (7) 

+(B+2Cy+&c.)h      f  =  -j     +(A'+B'y+C'y*+&c.)h 
-f&c.  j       <•    +&c. 

As  this  equation  is  true  for  all  values  of  h,  the  coeffic- 
ients of  like  powers  of  h  are  severally  equal  $  277). 

The  coefficients  of  h°  are  identically  (§  37.  c)  the  same. 
Passing  then  to  the  coefficients  of  A1, 
B+2  Cy+ZDij 2+&c.  =  ^'+5^f  C'y 2+Z>y -f&c.  (8) 
Again,  as  this  equation  is  true  for  all  values  of  y,  the 
coefficients  of  like  powers  of  y  are  severally  equal  (§  277). 

.-.       B  —  A;  2C  =  B';  3B=  C> ;  ABzzzJD';  &c. 
or     B  —  A>;  G—\B';  D  =  \C> ;  E—\D;  &c.  (9) 
That  is, 

a.)  B  is  found  by  substituting  x+h  for  x  in  A  and  tak- 
ing the  coefficient  of  A1,  viz.  A'. 

*19 


222  BINOMIAL  THEOREM.  [§  294. 

Cis  found  by  substituting  x-\-h  for  x  in  B  (i.  e.  in  A1), 

and  taking  half  the  coefficient  of  A1,  viz.  \B'(=.\A"). 

See  §  292.  N.  4. 

D  is  found  by  substituting  x-\-h  for  x  in  C  (z=z  \B'  = 

J^t"),  and  taking  one  third  of  the  coefficient  of  h1,  viz. 

/       B"       A'"  \ 
%0'(=-— 0  =  9— 5)  ;     an(l    so   on-      Thus,   substituting 

these  values  of  B,  C,  &c.  in  (1)  of  §  288, 

A"        A'"  A1V 

(x+y)*  =  A+A>y+  —y  2+—  y5+  ^-^-f&c.     (10) 

b.)  Or,  in  otber  words,  B  is  the  first  derivate  of  A  ;  C 
is  half  the  first  derivate  of  B,  i.  e.  half  the  second  derivate 
of  A  (§  292.  N.  4) ;  D  is  one  third  of  the  first  derivate  of 
G,  i.  e.  one  sixth  of  the  second  derivate  of  B,  i.  e.  one  sixth 
of  the  third  derivate  of  A ;  and  so  on. 

§  294.   Now  we  have  (§§  290)  292) 

A  =  xn ;  and  Ax+h  =  (x-\-h)n ; 
or  A-\-A'h-\-&c.  =  xn+nxn-lh+&c, 

A'zrznx"-*.  §277. 

B{=.A!)  =  nxn-^i 
as  we  found  it  before  (§  290). 

In  like  manner,  Bx+h  =  n(x-\-h)n-1  ; 

or  B-\-B'h+&c.  =  nxn~x-\-n{n— l)xn-2h+&c. 

B'  =  n(n—l)xn-2.  §277, 

C{  =  \B<  =  \A')  =  5&±LV*. 

_n(n—l)t 
1.2 


So  Cx+h=-±^(x+h)»-*  ;  or 


n(n — 1)       „  .  n(n — l)(n — 2)   „  ._  ,  , 

J  .  _  j 

..     X>(_}0   -2<3^   -2.3"1   '  1.2.3 

&c. 


§  295.]  DERIVATIVE. — COEFFICIENTS.  223 

a.)  It  will  not  be  necessary  actually  to  make  the  substi- 
tution of  x-\-7i  for  x.  For  the  coefficient  of  each  power  of 
y  is  of  the  form  Mxn ;  and  the  two  first  terms  of  M(x-\-K)n 
are  Mxn-\-Mnxn~1h  (§  290).     Hence, 

The  derivative  of  each  coefficient  of  y  in  the  series  is 
found  by  multiplying  that  coefficient  by  the  exponent  of  x, 
and  diminishing  that  exponent  by  unity.     Therefore, 

To  find  the  coefficient  of  any  power  of  y, 

§295.  b.)  Multiply  the  coefficient  of  the  preceding 
term  by  the  exponent  of  x  in  that  term,  diminish  the 
exponent  by  unity,  and  divide  by  the  number  of  terms 
preceding  the  required  term. 

Thus,  the  coefficient  of  yp  (i.  e.  of  the  term  which  has  p 

terms  before  it)  will  be  — - — -^    0    ' — - — £-- — J-xn-p.  (12) 

1.2.3  .  .  .  .  p 

c.)  The  term  -i 7;        ' 1 — LJLJx»-r.yP 

1.2.6 p  J 

is  called  the  general  term  of  the  series ;  because  if  we  make 
in  it  p  =  1,  2,  3,  &c,  successively,  we  shall  have  the  cor- 
responding terms  of  the  series. 

d.)  If  n  is  a  positive  integer,  the  series  tvill  terminate,  as 
we  have  seen  (§  285.  a).  But,  if  n  is  negative,  the  subtrac- 
tion of  unity  will  numerically  increase  the  exponent  with- 
out limit ;  and,  if  n  is  a  positive  fraction,  the  subtraction 
of  whole  units  will  first  render  the  exponent  negative,  and 
then  numerically  increase  it  in  like  manner.  Hence,  if  n 
be  either  negative,  or  fractional,  the  series  will  be  infinite. 

e.)   The  sum  of  the  exponents  of  x  and  y  in  each  term 
is,  evidently,  constant,  and  always  equal  to  n. 
f)  If  — y  be  substituted  for  +y,the  terms  containing  the 


224  BINOMIAL  THEOREM.  [§  296. 

odd  powers  of  y  will,  of  course,  be  negative  (§  237).     Thus, 
(x—y)l  =  a»—na?-1g+  n^~^xn-*y*-&c.     (13) 

g.)   Let  se  =  l,  and  y  =  1,  in  formula  (11).     Then 
,  rcfn— 1)   ,  w(re— l)(rc— 2)  .  . 

Hence,  in  an_y  poiver  whatever  of  the  si«n  of  two  quanti- 
ties, the  sum  of  the  coefficients  is  equal  to  that  same  power 
of  2- 

h.)   Let  x  =  l,  and  y  =  l,  in  formula  (13).     Then 

,   n(n— 1)      w(w— l)(w— 2j  .  . 
( 1-1)"  =  0  =  l-n+  A_J j-^-  +&c. 

Hence,  in  any  power  whatever  of  the  difference  of  two 
quantities,  the  sum  of  the  coefficients  is  equal  to  zero;  i.  e. 
the  sum  of  the  positive,  is  equal  to  the  sum  of  the  negative 
coefficients. 

i.)  Let  n  ■=■-.     Then 


=  as±^~Vf^r^»~V±&c-  (14) 

g  1  .  z  .  q 

-u.i-py  i  pip— 9)  Y~  ±p(p—q)(p—2<i)  y3  ■ 

"^'l         ja:1"    1.2j»a!s  1.2.3     ?3      re3"1" 

P(p-^)(p-2g)(y-8g)y*+jr    ?  (U) 

1.2.3.4         <?*        x4  $' 


§  296.   1.   (a-\-x)~l  =  what  ?     See  §  87.  c.  1. 

Here  «  =  — 1. 

(«-|-a;)-i  —  a-1— a~2a;+a-3a:2— a~4a;3+&c. 

=  «-i(l— a-xa;+a-2a;2— &c.) 

=i(l_5+5!_Ao.). 

a\        a      a3  / 


§  297.]  SERIES. — APPROXIMATION.  225 

2.  (a— x)-1  —  what  ?     See  §  87.  c.  2. 

3.  (1+tt2)"1  =  what  ?     See  §  87.  c.  4. 

4.  (a-j-z)-3  =  what?     See  §  87.  c.  5. 

5.  (a-fx)^rrwhat?     Compare  §§  173.  1 ;  281.  3. 
Arts.  a*-H*-*a4-  fc-«^2+  ^^^a^-ffcc. 

i  /        1  ar      Ik2.    1   x3         5    .r4  \ 

&C-  =  aA1+2«-8^  +  T6^-128^+&C-)- 

This  might  have  been  put  under  the  form 

and  the  last  form  of  the  answer  would  have  been  obtained  immedi- 
ately. The  following  examples  may  be  similarly  reduced.  It  is  well 
to  solve  them  under  both  forms. 

Also  let  a  =  100,  and  x  =  1 ;  a  =  400,  and  x  =  8 ;  &c 

6.  (a2+x2)^  =  what?     (i22— xrf?     See  §  173.  2. 

7.  s(a*—x*)s[=  (o»— a*)5  =  a^(l— ^)*J=what? 

3/         3   a;2        3   cc*        5   a?s        45   ^s  . 

Jn5.  „^1__  -__  ___  _-_  --&C.J. 

§  297.    1.   Extract  the  cube  root  of  65. 
65  =  64(1+^). 

.-.  (65)*=(M)*(l+rft)*=4(l+A)* 

ii        1    2 , 1  .  2      1    2    5  ,  1  x  3 

v  ~3    64 ~   1.2  V64/  ~     1.2.3     \64/  X 

=  4fl+— - + - &c.) 

\  ^3.26      3.Q.2l^^3.Q.9.2^  J 

=  4(1+  151  ~  36^64  +&C'^  *.020,724&c. 


226  BINOHIAL  THEOREM.  [§298. 

2.   What  is  the  tenth  root  of  1056(  =  102 4+32)  ? 
(1056)™  =  l024™(l+T§f3r)T*  =  2(1+ J^ 

\  ~10   32      200    V32/  ^6000V32/  / 

\  ~320      204,800^196,608,000  / 

§  298.  Let  a — b-\-c—f-\-g — h-\-k — Z+&c.  be  a  converging 
series  consisting  of  terms  alternately  ]^ositive  and  negative. 
It  is  required  to  determine  the  degree  of  approximation  at- 
tained when  we  stop  at  a  particular  term. 

If  we  stop  at  a  negative  term,  as  /,  there  will  remain  a 
set  o£  positive  quantities,  g — h,  h — ?,  &c.,  to  be  added  to  ob- 
tain the  true  sum  of  the  series.  Hence,  if  we  stop  at  a 
negative  term,  the  sum  of  the  terms  taken  is  too  small. 

1\,  on  the  other  hand,  we  stop  at  a  positive  term,  as  g, 
there  will  remain  a  set  of  negative  quantities,  — h-\-h, 
— /+;»,  &c,  to  be  added  to  obtain  the  true  sum  of  the  se- 
ries. Hence,  if  we  stop  at  a  positive  term,  the  sum  of  the 
terms  taken  is  too  great. 

Now  the  sum  of  the  terms,  before  g  was  added,  being  too 
small,  and,  after  g  was  added,  too  great,  the  error  in  the 
first  instance  must  have  been  less  than  g. 

In  the  same  manner,  it  is  evident,  that,  if  we  stop  at  g, 
the  error  is  numerically  less  than  h. 

Hence,  whatever  number  of  terms  of  a  converging  se- 
ries whose  terms  are  alternately  positive  and  negative  we 
take,  the  error  will  be  numerically  less  than  the  next  suc- 
ceeding term. 


CHAPTER  XIII. 


DIFFERENCES. 


§  299.  Let  there  be  given  the  series  of  square  numbers, 
1,  4,  9,  16,  25,  &c. 
If  now  we  subtract  the  first  of  these  numbers  from  the  se- 
cond, the  second  from  the  third,  &c,  we  shall  obtain  what 
is  called  the  first  order  of  differences.  If  then  we  subtract 
the  first  of  these  differences  from  the  second,  &c,  we  shall 
obtain  the  second  order  of  differences,  and  so  on. 
Thus,    1,  4,  9,  16,  25,  36,  49 

3,  5,  7,     9,    11,  13  the  first  order  of  differences. 
2,  2,    2,     2,     2,  second  "  «' 

0,  0,    0,     0,  third     "  " 

What  are  the  several  orders  of  differences  of  the  num- 
bers, 1,  4,  10,20,  35,  56,  &c? 

1,    4,    10,  20,  35,  56 
3,    6,    10,  15,  21  the  fir3t  order  of  difF. 
3,    4,      5,     6  second        " 

1,     1,     1  third  « 

0,     0  fourth         " 

§  300.    Let  there  be  an  increasing  series, 

a  ,  a2,  *a3,  a4,  &c.    Then  we  have 

a2 — ax,     a3 — a2,     a4 — a3,  &c,  first  order  of  difF. 
a3— 2a2+a1,     a4— 2a3+a2,  &c.,  second  " 
a4— 3a3+3a2— ax,  &c,         third  " 
&c. 


228  DIFFERENCES.  [§301. 

If  we  represent  the  first  terms  of  these  successive  orders 
by  D j,  D2,  D3,  Z>4,  &c.  we  shall  have 


Dx  —  a 


2      ai  > 


Dn--±a^na2±-^-^az^.         1£3     'a4±&c.;(2) 


D2  =  a3 — 2a2-\-a1 ; 

D3  =  a4 — 3a3-j-3a2 — ax  ; 

Z>4  =  «5— 4a4-f-6a3— 4a2-\-a1  ; 

and,  obviously, 

«(w— 1)  n(n— l)(n— 2)  , 

A  =  «„+!—««»+    1>2  «*-i 17273 — " a"- 2+ 

&c. ;      (1)  the  coefficients  of  an+van,  Sic.,  being  the 

coefficients  of  the  wth  power  of  a — x. 
Or,  reversing  the  order  of  the  terms, 

n(?i — 1)      ^?i(n — l)(w — 2) 

I 

taking  the  upper  signs  throughout,  when  n  is  even  ;  and  the 
lower  signs,  when  n  is  odd. 

Hence  the  first  terms  of  the  several  orders  of  differences 
may  be  found  without  finding  the  remaining  terms. 

1 .  What  is  the  first  term  of  the  third  order  of  differences 
of  the  series,  1,3,  6,  10,  15? 

Here  a1  =  l}  a2  •=.  3,  a3  z=.Q,  a^=z  10,  and  n  =  3. 

...    D3 (  =  a4— 3a3-f 3a2— xa)  =  10— 3X G+3X3— 1  =  0. 

So  we  should  have  D2  =:  ax — 2a2-\-a3  ==.  1 — 2x3-[-6  =  1. 

2.  Given  the  series  1,  8,  27,  64,  125,  to  find  B^JD^ 
D?  and  Z>4. 

Ans.  Dx  —  7,  D2  =  12,  D3  =  6,  and  Z>4  =  0. 

§  301.   From  the  values  of  Dlf  D„,  &c.  in  §  300  we  have 

a3  =  «1+22)1+Z)2, 

a4  =  a1+32>1+8Z>2»+i>s. 

and,  obviously, 

.  /       i^n    i  (w-l)(n-2)n 
a.=  al+(n— 1)D^ — D3 + 

(tt-l)(tt-2)'(n-3)n    ,  (n-l)(n-2)(n-3)n  . 

— ni — *»+ — r^3 — ^+&c-j  (2) 


§301.]  DIFFERENCE  SERIES.— ORDERS.  229 

the  coefficients  of  the  terms  being  the  coefficients  of  the 
(n — l)th  power  of  a-\-x. 

1.  What  is  the  fourth  term  of  the  series  of  squai'es, 

1,4,  9,  16,  &c.? 

Herea^l,   Dl=:3,   D2=i2,D5  —  0,   and  n  —  4. 

...  a4(=  fl1+3D1+3Z>2+D3)  =  1+3X 3+3x2+0  =  16. 

2.  What  is  the  twentieth  term  of  the  same  series  ? 

Ans.  400. 

3.  What  is.  the  nth  term  of  the  same  series  ? 

an=al+(n-l)D1+  MK^j^ 

==  l+3(?*— l)+(w— 1)  (n— 2). 

==  1+3  (»— !)+«(«— 1)— 2  (n— 1). 

=  1+m — l+?z(n — 1)  =n2. 

4.  What  is  the  nth  term  of  the  series, 

a,  a+D,  a+2Z>,  «+3Z>,  &c.  ? 

^4ns.  «+(n — 1)Z>, 

Notes.  (1.)  The  problem  contained  in  the  last  example  has  been 
already  considered  (§250).  In  fact,  the  whole  subject  of  equidiffe- 
rent  series,  there  treated,  is  only  a  particular  case  of  the  more  gene- 
ral subject  of  differences  ;  viz.  the  case,  in  which  the  first  differences 
are  constant  (§  249);  and,  of  course,  the  second,  and  all  higher  dif- 
ferences are  equal  to  zero. 

(2.)  It  is  proper  to  remark  here,  that  an  equidifferent  series,  hav- 
ing its  first  differences  constant,  is  called  a  difference  series  of  the 
first  order  ;  a  series  whose  second  differences  are  constant,  is  said  to 
be  of  the  second  order;  and  so  on.     Thus,  we  have  the  series, 

1,  2,  3,  4,  5,  6,  of  the  first  order. 

1,  3,  6,  10,  15,  21,  second." 

1,  4,  10,  20,  35,  56,  third      " 

(3.)  These,  which  are  only  particular  examples  of  the  various  or- 
ders of  difference  series,  have  also  this  property;    viz.  the  nth  term 
of  each  series  is  equal  to  the  (« — l)th  term  of  the  same  series  plus 
the  nth  term  of  the  preceding  series.     And,  consequently,  the  nth 
ALG.  20 


- 

■ 


N.-r  : 

sr  vcmfcas:  teemse  tbe  ■■iirtu   «  saberical  bodies,  at 

fe»  ic  tr  ««;  tiara-.  the  naber  of  balfe  ea  eacb  ale 

>«ns  ejy,u«.d  bj  tke  urn  yn—TiL.  torn  of  tbe  aatnal  series 

-?tbenaili      rffe  riesaracaMa^rosMfeJ 

"--   -  i  J     -  -  —  -  ~  .:  :^ '"-   : :    ..-.-"...   ::   \-'-a:r:z     :ir    r  - :    r  -::  ; 
Wt  ia  tbe  fewest  l— ui.i  boaf  expressed  by  tbe  tant-ydag  Sena 

Wii:  is  li*  fifteenth  term  of  the  ser 

Am 
*.   Wbii  is  tbe  *±  term  of  the  se:  ?.  €,  1» 

-- 


J    f. 


—      - 


"     ITIju  iszlxMik  torn  oe 

-:     — . 

; . 

—  -         -  __;_      —    Sc: 

=  .  -         •  •     -  • 

-   - 

7;;  :=."  ;    "       •■-.—:;:::  :  -  :  — '. 

r_r  i--  .-h?  ::"   fn.:i  r'-r  :.  \- .  '-1  :  — I.rad — 2ad  -  «■?*". -h  : 

fraud  to  be  eaek  ercsl  to  0.     Ttos  wiB  be  9- 
-VH.r  tbe  saceesaea  ef  difieraKce  :' as  ef  die  I 
t  -- 

7 1  :.i 


-      --    —I,      •,      I,      «»      £»,     4  5,]        «, 

l.     i,     l.  1,3      i» 


§  302.]  INTERPOLATION.  23 1 

In  the  corresponding  series  of  the  fourth  order,  1,  5,  15,  35,  70, 
126,  we  should  find  four  terms  equal  to  zero,  and  the  terms,  corres- 
ponding to  the  negative  local-  indices  beyond,  positive;  and  so  on,  t'ie 
number  of  terms  each  equal  to  zero  being  equal  to  the  number  of  the 
order ;  and  the  terms,  corresponding  to  the  negative  indices  beyond, 
being  positive  or  negative  according  as  the  number  of  the  order  is 
even  or  odd. 

§  302.  The  formula  (2)  of  the  preceding  section  had  pri- 
mary reference  to  those  terms  only  whose  place  in  the  se- 
ries is  expressed  by  whole  numbers ;  i.  e.  to  those  denoted 
by  integral  local  indices.  *\Yq  have  found,  however,  by 
taking  ra  =  ^,  f,  &c.  in  the  general  solution  of  examples 
Cth  and  7th,  terms  corresponding  to  those  fractional  local 
indices,  and  still  conforming  to  the  general  law  of  tile- 
ries. 

"We  shall  find,  in  like  manner,  that  the  above  formula 
applies  in  general  to  such  intermediate  terms  correspond- 
ing to  fractional  local  indices,  equally  as  to  terms  whose 
local  indices  are  integral ;  only  giving  a  suitable  value  to 
n  (§  263). 

Note.  This  is  simply  a  more  •general  form  of  the  problem  of  in- 
terpolation ;  and  applies  to  all  series,  whose  differences  of  any  or- 
der become  either  zero,  or  so  small  that  they  may  be  neglected. 

1.  Given  22  =  4,  32  =  9,  and  42  =  16  ;   to  find  (2£)*. 
Here  a1  =4:,D1~  5,  D2  —  2,  D3  =  Oj  and  n  —  it 

an  =  a^  =  4  +  1X0  —  $X2  =  6£. 

2.  Given     (2500)*  =  50,       (2501)*  =  50.009,999,8, 

(2502)*  =50.019,999,6;  to  find  (2500.5)*. 

Here  a1  =  50,  D1  =  .009,999,8,  Da  =  0  ;   and  n  ==  1> . 

.-.  an=  (2500.5)*  =  50-Hrx.009,999,8  =  50.004,999,9&c. 

3.  Given  64*  =  8,      66*=  8.124,038, 

68*  =  8.246,211,  and  70*=  8.3666;  to  find  65*. 


(z)    Lat.  locus,  place.     Local  indices  indicate  the  place  of  the 
term  in  the  series. 


232  DIFFERENCES.  [§  303. 

Here       at=8,   D^  =.124,038,   D2  =  —  .001,865, 
&3  =  J(.000,076-i-.000,081)  =  .000,078 ;   and  n  =  l£. 

Ans.  (65)^  =  8.062,257. 

4.  Interpolate  3  terms  between  the  fourth  and  fifth 
terms  of  the  series, 

4,  8,  12,  16,  20. 

Here  ax  =  4,  Dx  =  4,  D2  =  0 ;  and  n  =  4^,  4£,  4|, 
«n=4  +  3iX4  =  17;  &c. 

Or  «x  =  16,  Dx  =  4,  D2  =  0  ;  and  »  =  1J,  If,  If. 
«?1=16  +  1X4  =  17;  &c. 

Ans.  17,  18  and  19.     See  §  263. 

5.  We  find,  in  a  table  of  natural  sines, 

sin  30=  =  .5,  sin  30=  10'  =  .502,517, 

sin  30=  207  =  .505,030,     sin  30°  30'  =  .507,538. 

What  is  the  sine  of  30°  1'  ?  of  30°  2'  ?  of  30°,  3'  ? 
of  30'  4'  ? 

Ans.  sin  30=  V  =  .500,252  ;     sin  30=  2'  =  .500,504  ; 
sin  30=  3'  =  .500,75  6  ;     sin  30=  4'  =  .501,007. 

§  303.  a.)  In  finding  a  term  of  the  series  by  §  301,  n 
being  a  whole  number,  the  formula  (2)  will  always  termi- 
nate, because  the  coefficient  n(n — 1) (?i — ?i)  =  0, 

But,  in  interpolation  (§  302),  the  formula  will  not  terminate, 
unless  we  find  an  order  of  differences  equal  to  zero.  For 
n  being  fractional,  none  of  the  factors,  n — 1,  n — 2,  &c,  can 
become  zero  ;  but  they  will  become  negative,  and  then  in- 
crease numerically  (§  295.  d).  Iii  this  case,  the  required 
term  can  be  found  only  by  an  infinite  series. 

b.)  It  will  have  been  observed,  that  Ave  have  found  terms, 
whose  places  are  expressed  both  by  integral  and  by  frac- 
tional local  indices,  without  knowing  the  law  of  the  series 
into  which  they  are  introduced  ;  knowing,  in  fact,  nothing 
of  the  series  but  a  few  terms  ;  or  even  a  single  term  with 
the  successive  differences. 


§304.]  sum.  i 

c.)  Hence,  obviously,  the  differences,  together  with  u 
single  term,  determine  the  character  of  the  series.  Tht-v 
enable  us  to  continue  the  series  to  any  extent  (§  301),  to 
supply  intermediate  terms  (§  302),  and,  as  we  shall  sec 
(§  304),  to  find  the  sum  of  any  number  of  terms. 

§304.  Let  it  be  required  to  find  the  sum  of  n  term-  oi 
the  series, 

flj,      Ct2)      Cls,      ft4,      655,    ....    Ctn. 

Assume  a  series,  whose  first  differences  shall  be  the 
terms  of  the  given  series.     Thus, 

0,    ctj,    ax-\-a2,    a1-{-a2Jj-a3,  .  .  ax-\-a2-\-az  .  .  -f-c*,,. 

Now  the  (?j-(-l)th  term  of  this  last  series  is,,  evidently . 
the  sum  of  n  terms  of  the  given  series;  and  the  (n-\-l)th 
differences  of  the  last  series  are  the  nth.  differences  of  the 
given  series. 

Hence,  marking  the  terms  and  differences  of  the  assume 
series  with  the  accent ',  we  have,  in  formula  (2)  of  §  301, 

a?1  =  0,   D\—ax,   D'z  =  I)1,&,c.; 
and,  putting  7i-\~l  in  place  of  n,  and  denoting  by  S  the  re- 
quired sum  of  n  terms  of  the  given  series  (i.  e.  the  (n-\~l)tL 
term,  ci'n+1,  of  the  assumed  series),  we  find 

5(  =  „<„+I)  =na1+  ^i>1+"("71»72)Oa+ 

1 .  What  is  the  sum  of  n  terms  of  the  series, 

1,  2,  3,  4,  5,  G,  &c.  ? 

Herea1  =  l,    Dl=zl,    and  D2=z0. 

■      n(n—l)      n(»+l) 
b  =  n-\ — ^ — — -=— —^ — -.     See  §  2o(j.  3. 
1      1.2  2 

2.  "What  is  the  sum  of  n  terms  of  the  series,  a,  a-\-D, 
a-\-2D,  &c.  (i.  e.  an  equidifferent  series)? 

Ans.  na+\n(n—l)D.     See  §§  253 ;  301.  N.  1. 

*20 


234  DIFFERENCES.  [§  304. 

3.  What  is  the  sum  of  n  terms  of  the  series, 

1,  3,  G,  10,  15,  21,  28,  &c.  ? 
Here  ax  =  1,D1  =  2,  D2  =  1,  and  D3  =  0. 

a         .     t       in  ■  n(n-l)(n-2)  _n(n+l)(n+2) 
...     S=n+n(n-l)^ r0—       1<2   3       . 

4.  What  is  the  sum  of  «  terms  of  the  series, 

1,  4,  10,  20,  35,  &c.  ? 

n(w+l)(n+2)(n+3) 
^nS*  1.2.3.4         ' 

5.  What  is  the  sum  of  n  terms  of  the  series, 

1,  3,  5,  7,  9,  11,  13,  &c  ? 

Ans.  n2.     See  §256.  5. 

*?.   What  is  the  sum  of  n  terms  of  the  series, 

12,  2*,  3*,  42,  5 2,  &c? 

w(n+l)(2H-l) 
'4nS-         17273        ' 

7.   What  is  the  sum  of  n  terms  of  the  series, 

13,  23,  33,  43,  53,  &c? 


j^«wv=(&*£y. 


Notb.    From  the  result  of  examples  1st  and  7th,  we  have 
13  +  23+33  ..+713  =  (1  +  2  +  3  ..+*)*. 


CHAPTER  XIV. 


INFINITE  SEEIES. 


§  305.  An  infinite  series,  we  have  seen,  may  arise 
from  an  imperfect  division  (§  87.  a) ;  or  from  the  extraction 
of  a  root  of  an  imperfect  power  (§  170.  N.  5) ;  or  by  the 
continuation  of  an  equimultiple  (§  261.  c)  series  to  infinity* 

Infinite  series  of  various  forms  are  also  developed  by  the 
method  of  undetermined  coefficients  (§  281),  and  by  the  bi- 
nomial Theorem  (§  295.  d) ;  and  by  many  other  processes, 
which  we  are  not  yet  prepared  to  investigate,  and  some  of 
which  are  beyond  the  reach  of  elementary  Algebra. 

§  306.  As  the  processes  of  developing  infinite  series  are 
so  various,  the  methods  of  summing  them  are  equally  vari- 
ous. Even  of  those  which  are  summed  by  the  elementary 
processes  of  Algebra,  we  shall  consider  here  only  one  or 
two  of  the  simplest. 

a.)  The  method  of  summing  a  converging  infinite  equi- 
multiple series  has  already  been  investigated  (§  261.  c). 

b.)  The  true  sum  of  an  infinite  series  resulting  from  di- 
vision, or  from  the  development  of  a  fraction  by  undeter- 
mined coefficients,  is  the  fraction  from  whose  development 
the  series  originated ;  and  this,  whether  the  series  be  con- 
verging or  diverging  (§  87.  d.f). 

We  may,  moreover,  approximate  to  the  value  of  a  con- 
verging series  by  the  actual  addition  of  a  small  number  of 


236  INFINITE  SERIES.  [§  307. 

the  terms  (smaller  or  greatei',  according  to  the  greater  or 
less  rapidity  of  the  convergence). 

But  the  doctrine  of  infinite  series  proposes  to  find  con- 
venient expressions  for  the  sum  of  any  part,  or  the  whole 
of  a  series,  without  the  labor  of  adding  the  several  terms. 

§307.   We  have    ■£- 7- =  -7 ~Vv  §  118- 

.-.  (§42.  d)  _4— =!(£ £-).     (1)    Thatis, 

A  fraction  of  the  form  — ; — ; — -  is  equal  to  -  of  the  dif- 

m{m-\-p)  p 

fcrence  between  the  two  fractions  —  and  — ; — .     Now,  as 
J  m  m-j-p 

this  is  true  of  any  fraction  of  this  form,  it  is^true  of  each  of 

the  terms  of  a  series  composed  of  such  fractions.     Hence 

the  sum  of  such  a  series  will  be  equal  to  -  of  the  difference 
between  two  series,  one  consisting  of  terms  of  the  form  — , 

fit 

9 


and  the  other,  of  the  form 

m-\-p 

I.   Let  it  be  required  to  find  the  sum  of  the  series, 

1 1 h&c,  to  n  terms,  and  also  to  infinity. 

1.2~2.3      o.4^ 

Here  we  have 

m{m+p)  =1.2,     2.3,  &c.  =  l(l-j-l),  2(2+1),  &c. 
.-.     q=z  1,    p=zl,    and  m  =  1,  2.  3  .  .  n,    successive  ly. 

Represent  also  the  sum  of  n  terms  by  Sn,  and,  by 
gy,  the  sum  of  an  infinite  number  of  terms  by  S^.     Taen 


n  1  —  1 == . 


1  if"  «+l     ~ "+1 

s" 


_i    *  * '  '  "    n      n+1  J 


If  n  =  00,  we  have  (§  138)  —r-r  =  0  ;  and  .-.  Sa>  =  1. 


§  307.] 


SUMMATION. 


Otherwise;  when  n  =  cc,  we  have  (§  261. 
1  ■  =  -  =  !;     and  .•.£„=! 


n-\-\       n 
1.   What  i3  the  sum  of  the  series,  - — -  -\-  — -  -f-  — --  -f- 

1  .  c>  O  .  O  Hit 

&c.,  to  il  terms,  and  also  to  infinity  ? 

IIere  o(=i(i^)'  o(-3^T2))'  &c'are  of  t!l! 

form 


(2n-l>[(2n-l)-r-2]- 

We  have,  therefore  ^  —  1,  j*>  :=  2,  and  m  r=  2>i — 1 . 


Hence,  Sn—-< 


1+1+*    •  •••  + 


3  '  5 


2n— 1 
1 


[        3      5'"''       2«— 1       2n+l  J 

<?  —±h       1 "  >  —    ?* 

rt  — 2  V1     2n+l)~  2n+l' 
Also,  making  ?j  z=  oo,  we  have 

n  n  n  t         r» 

=  °>  or  orr-r  =  nr ;   and  .-.  >S'O0  —  - 


2«-J-l 


2w-hl       2?i ' 


CO  —  r> 


8.   Find  the  sum  of  n  terms  of  the  series, 


+J 


1      ,     1      ,  » 
+  _+&c; 


1.4  '  2.5  '  3.6  '  4.7 
and  also  of  the  whole  series  to  infinity 

HereIoW       2(2^3)'  &C-  Sive 
q  =  1, p  —  3,  and  rani,  2,  3 


n. 


„       lf+2  +  3+I-+» 
^«  —  *i  l  i 


°  i 


1     i 


> 


! 


n      n-\-\       n-\-2       n-f-3  J 


238  INFINITE  SERIES.  [§  307. 

S  -  Vi  +  i  +  i I I ±J\ 

"3\  r2"r3   w+1  n+2      n+3J' 


s( 


w-j-1      «+2      rc+3 , 
n  n       .       n 


re+1    '  2n+4   '  3w+9. 

n       ,         /i1  w 


3w+3    '  6^+12    '  9»+27 

AIM :fl.S=i(l+H-*)  =  H- 

4.  Find  the  sum  of  the  series,  1+H~^4-tV+&c-  5  (the 
denominators  being  the  terms  of  a  differential  series  of  the 
seeond  order,  viz.  the  triangular  numbers ;). 

Dividing  the  series  by  2,  we  have 

by  example  1st,  above. 

S      -  2 


o.    "What  is  the  sum  of  the  series, 
1  1.1 


+&c. 


3.8  '   6.12  '  9.16 

Take  out  of  each  term  the  common  factor  ^,  by  divid- 
ing the  second  factors  of  the  denominator  by  4,  and  the 
first  factors  by  3.  as  * 

6.    Sum  the  series,  A  _  _  +  _  _  _  +&c. 

Here  p  —  2,q  —  n-fl,  and  m  —  2ra+l ;  n  being  taken 
=  1,  2,  3,  &c.j  successively. 

\"2      3      4      5  w-fl  1 

-iis-^-^- "  ,:f2»+i 

•'•     ^»  — 2]       _2      3      4  w  w+1    (' 

I         5  +  7      O"1"'  '  'T2n+1   '   2?i+3j 

Note.    If  n  is  infinite, 

i-i+i-i+&c.=q^=*. 


§  308.]  SUMMATION.  239 

The  sum  of  n  terms  of  this  series  will  be  equal  to  1,  if  n  is  finite 
and  even ;  to  0,  if  it  is  finite  and  odd.     Hence  we  have 

S0B=*[§-(i-i+i-i+&c.)]  =  Ki-i)  =  tV 

_1/2_1       1       n+1  x    .1(1       /1_    n±l_\} 
n~ 2V3      2  T  2      2w+3  /      2  1  G  T  V2      2re+3/  j  ' 

_1/1      L-Wf-L      *       > 

~2VG:F2(27i+3)y'       Vl2:F4(2rt+3)>/• 

7.   What  is  the  sum  of  the  series, 

Am.  S„  =  -i-^x  +  ~);  S„=j. 

'  8.   What  is  the  sum  of  the  series, 
1  1.1 


— —  +  ~  — &c.  ? 

1.3      2.4^3.5 


Am.  S^  =  -. 


9.   What  is  the  sum  of  the  series, 

1.5^5.9^9.13^13.17^  * 

Here  q  =  4,  p  =  4,  and  m  =  4»+l;  «  being  taken  =0,  1,  2, 
3,  &c,  successively. 


rp  t 


§  308.    Again,  as  we  evidently  have  (§  118) 

q  9 


I 


p  i  m(m-\-p)  ..  [m-\-(r—l)p)      (m+p)  {m+2p) . .  {m+rp) 

I ,  (2) 

m{m-\-p){m-\-2p)   .  .  .  (wi+rp)' 

a  series  of  terms  in  the  form  of  the  second  member  of 

this  equation  is  equal  to  —  of  the  difference  of  two  serie3 

of  terms  in  the  form  of  those  in  the  second  factor  of  the 
first  member. 


240  INFINITE  SERIES.  [§  308. 

A  K  C 

1.   Sum  the  series,  —- ^ .+  ^-^.+r— g  +&c. 

Here  <?  =  ?z-}-3,  ^  =  1,  r  =  2,  and  ??*  =  n  —  1,  2,  3,  &c. 


(i±2  +  2L3  +  31i+&C')4by§307-1- 


1/  4 

2 


2.    Given  _*-,  +j^  +  ^  +&C  to  find  5. . 
3'    GireniT2^  +  2^^  +  oT4i5-^+&C-'t0 


4.    Given,    _    „    „-4-a    g    ~   a-+c   r?    „-<:+  &c-> 


1 
1T3T5T7  "^  3T57T79  n~  5T7T9T1 1 

5'    Giveni-2^r4  +  2-^^  +  3T4^'t0   fmd 
S».  ,       «    -89 

^In*.  /Sao  —  gj> 


CHAPTER   XV. 


LOGARITHMS. 


§  309.  All  finite,  positive  numbers  may  be  regard- 
ed as  powers  of  any  finite,  positive  number  except 
unity. 

Thus,  if  10  be  taken  as  the  base  (§  22.  N.),  1,  10,  100, 
1000,  &c,  xVs  xfoy>  t^Vo'  &c->  Wl^  De  expressed  as  integral 
powers  of  the  base;  those  above  1, positive ;  those  below, 
negative. 

Moreover,  it  is  obvious,  that  all  numbers  between  the  in- 
tegral powers  can  be  expressed  as  fractional  powers,  either 
positive  or  negative.  That  is,  the  base  can  be  separated 
into  factors  so  small,  that^  a  certain  number  of  them  multi- 
plied together  (§  12),  or  divided  out  of  unity  (§  14),  shall 
produce,  at  least  to  any  degree  of  approximation,  any  given 
number  ($319). 

a.)  It  is  evident  that  1  cannot  be  taken  as  a  base  of  such 
a  system  of  powers,  because  every  power  of  1  is  1. 

b.)  It  is  also  evident,  that,  if  a  proper  fraction,  as  y1^,  be 
taken  for  the  base,  fractions  will  be  expressed  as  positive, 
and  integers,  as  negative  powers. 

c.)   The  base  must  be  a  positive  number ;   for  if  it  were 
negative,  only  such  positive  numbers  could  be  expressed  as 
should  coincide  with  its  even  powers ;  and  only  such  nega- 
tive numbers,  as  should  coincide  with  its  odd  powers. 
alg.  21 


242  LOGARITHMS.  [§310-312. 

d.)  Again,  of  a  positive  base  no  negative  number  can  be 
a  power,  unless  the  denominator  of  its  exponent  be  even,, 
and  the  numerator  odd  (§§  11.  N.  2 ;  23.  e,f).  Hence  the 
limitation  to  positive  numbers. 

§  310.  If  all  numbers,  with  the  limitations  above 
explained,  were  thus  expressed  as  powers  of  a  sin- 
gle number,  the  labor  of  multiplication  and  division 
would  obviously  be  reduced  to  the  adding  and  sub- 
tracting of  the  exponents  (§§  15,  16). 

Thus,  since  100  =  102,  and  1000  =  103  ; 

100X 1000  =  102X103  =  105  =  100  000. 

Also,  iooo  =  io;>,    Ti¥  =  io-2. 

10(KH-Tio  =  103-M0-2  —  105  —  ioo  000. 

8  01  0  30  JL9897 ^O 

2-=z\Ql'0O0'S'U0  ft  —  i()T  oo  OVISTS^ 

•2x5  =  10-3olosoXl0-G*S07O  =  10l. 

)  311.  When  numbers  are  thus  expressed  as  pow- 

of  another  number,  the  exponents  of  those  powers 

are  called  logarithms"  of  the  numbers  so  expressed; 

and  the  number  whose  powers  are  thus  employed,  is 

ed  the  base  (§22.  N.)f  and   sometimes  also  the 

radix  (§23.  d),  of  the  system. 

r.  fence,  for  a  given  base, 

§  312.  The  logarithm  of  any  number  is  the  expon- 
ent of  the  power  to  which  the  base  must  be  raised,  to 
produce  that  number. 

Tims',  2  is  the  logarithm  of  100  to  the  base  10 ;  because 
;  the  exponent  of  the  power  to  which  10  must  be  raised 
to  produce  100. 

So,  because  2  =  10-301030,  .301030  is  the  logarithm  of 
>  the  base  10  ;  for  .301  030  is  the  exponent  of  the  pow- 
er to  which  the  base,  10,  must  be  raised  to  produce  2. 
t)  dr.  ?oy»r,  ratio,  &pt&ftbg,  number ;  number  of  the  ratio. 


§313,314.]  CHARACTERISTIC.  243 

or.)  Cor.  I.  The  logarithm  of  the  base  of  the  system  is  al- 
ways 1.     §  11.  a. 

b.)  Cor.  II.  The  logarithm  of  unity  in  every  system  is 
zero.     §  13. 

§313.  The  base  of  the  system  of  logarithms  in  common 
use  is  10.     "We  have,  therefore, 

log  10,000  =  log  104  =  4,    log  1000  =  log  10^  —  3, 

log  (100  =  10 2)=:  2,       log  (10  =  lO1)^, 

log(l=10°)  =  0,     log  (TV  =  10-i)  =_1} 

log  (.01  =  10-2)  =  —2,      log  (.001  =  10-3)  _  _3j  &c. 

a.)  Hence,  obviously,  the  common  logarithm  of  any 
number  between  1  and  10  is  a  proper  fraction ;  that  of 
any  number  between  10  and  100  is  1  -f-  a  fraction;  be- 
tween 100  and  1000,  it  is  2  +  a  fraction  ;  &c. 

b.)  Again,  the  common  logarithm  of  any  number  between 
■fo  and  1,  as  .3454,  is  between  —1  and  0,  and  therefore  it 
is  — 1  -\-  a  fraction ;  of  a  number  between  .01  and  .1,  as 
.0205,  the  logarithm  is  —2  -j-  a  fraction ;  of  a  number  be- 
tween .001  and  .01,  the  logarithm  is  —3  -f-  a  fraction. 

§  314.  c.)  The  integral  part  of  a  common  logarithm  is 
called  its  characteristic  ;  because  it  characterizes  the 
logarithm  by  showing,  where  in  the  series  of  the  powers  of 
10  the  number  of  which  it  is  the  logarithm  falls.  TIio 
characteristic  of  the  logarithm  of  a  number  greater  than 
ten  is  positive;  of  a  number  less  than  unity,  negatire 
(§  309). 

d.)  Moreover  (§313),  the  characteristic  of  the  common 
logarithm  of  any  number  is  always  equal  to  tho  exponent 
of  the  integral  poioer  of  10  next  below  that  number;  and 
hence,  in  the  common  system, 

(1.)  If  a  number  be  greater  than  unity,  the  characteristic 
of  its  logarithm  is  one  less  than  the  number  of  its  integral 
places;  (2.)  if  less  than  unity,  the  negative  eharacteristii 


244  LOGARITHMS.  [§315,316. 

is  numerically  one  greater  than  the  number  of  cyphers  be- 
tween the  decimal  point  and  the  first  significant  figure  on 
the  left,  in  the  decimal  expression  of  the  fraction. 

e.)  Otherwise;  the  characteristic  of  a  logarithm  of  a 
number  is  equal  to  the  number  of  places  from  the  unit  place 
to  the  highest  significant  figure,  including  the  latter  ;  posi- 
tive, if  that  figure  be  on  the  left  of  the  unit  place ;  negative, 
if  on  the  right. 

§  315.  /.)  We  have  log  20  =  log  10  +  log  2  =  1  +  log  2 : 

log200=log  100  +  log2  =  2  +  log  2; 

log  525  =  log  10  +  log  52.5  =  1  +  log  52.5  ; 

=  log  100  +  log  5.25  =  2  +  log  5.25. 

So     log  .525  =  log  525  —  log  1000  =  —  3  +  log  525. 

But  adding  whole  units  to  a  mixed  number  cannot  afi'ec^ 
its  fractional  part.  Hence,  the  decimal  part  of  the  com- 
mon logarithm  corresponding  to  a  number  expressed  by 
any  figures  whatever,  is  the  same,  whether  those  figures 
stand  all  on  the  right,  or  a  part  or  all  on  the  left  of  the  dec- 
imal point.     Thus,  we  have 

log  25  =  1.397  960  ;     log  250  =  2.307  9G0  ; 

log  25  000  =  4.397  9G0  ;    log  .025  =—2.397.960. 

g.)  The  principles  of  §§  313-315  result  from  the  employ- 
ment of  the  base  of  our  scale  of  notation  as  base  of  the  sys- 
tem of  logarithms.  *  On  account  of  this  peculiarity,  the 
common,  or  Briggs's1  logarithms  are  much  more  convenient 
than  any  other  for  numerical  computations  ;  and  are,  there- 
lore,  in  universal  use  for  that  purpose. 

§310.  The  following  principles,  resulting  from  the  na- 
ture of  logarithms  as  exponents  (§§  309-312),  are  formally 
stated  here  for  reference. 


(&)  So  called  from  Mr.  Henry  Biiggs,  who  first  suggested  to  Lord 
Napier,  the  inventor  of  logarithms,  the  employment  of  10  as  a  base; 
and  who  completed  the  computation  of  the  first  table  of  logarithms 
\vith  that  base. 


§317,319.]  INTERPOLATION.  245 

1.  The  sum  of  the  logarithms  of  any  number  of  fact 
is  equal  to  the  logarithm  of  their  product  {§  15). 

2.  The  logarithm  of  a  dividend  minus  the  logarithm  of  a 
divisor,  is  equal  to  the  logarithm  of  their  quotient  (§  16  . 

3.  The  double  of  the  logarithm  of  a  number  is  equal  to 
the  logarithm  of  the  square  of  that  number ;  the  triple  of  ids 
logarithm,  to  the  logarithm  of  its  cube,  &c. ;  the  half  to  the 
logarithm  of  its  square  root;  one  third,  to  the  logarithm  of 
its  cube  root,  &c. ;  and,  in  general,  n  times  the  logarithm  •</ 
a  number  is  equal  to  the  logarithm  of  the  nth  power  of  the 
number  (whether  n  be  integral  or  fractional,  positive  or 
negative).     See  §  24.  d. 

§  317.  It  is  evident,  that  if  a  set  of  numbers  form  an 
equimultiple  (§  257)  series,  their  logarithms  will  form  an 
equidifferent  (§  249)  series. 

Thus,  the  logarithms  of  1,  10,  100,  1000,  are  0,  1,  2,  3. 

So  the  logarithms  of  a,  am,  am"2,  &c.,  form  an  equidiffe- 
rent series,  of  which  the  common  difference  is  the  logarithm 
of/N.     Hence, 

§  318.  If  between  two  numbers  we  interpolate  any  num- 
ber of  equimultiple  means  (§  265),  and  between  the  corres 
ponding  logarithms  interpolate  the  same  number  of  equi- 
diiferent  means  (§  255),  these  last  terms  will  form  the  log- 
arithms of  the  several  terms  of  the  first  series.     Thus, 

The  equimultiple  mean  between  1  and  10  =  3.162  277  7. 

The  equidifferent  mean  between  0  and  1  =  \. 

102  3.162  277  7  =i. 


'o 


§  319.  If  the  base,  10,  be  separated  into  1  000  000  equal 
factors,  301  030  of  these  factors  multiplied  together  wil*. 
within  an  extremely  small  fraction,  produce  2;  in  like 
manner,  477  121  will  produce  3  ;  602  060  will  produce  4  ; 
500  000  will  produce  3.162  277  7  ;  and  so  on.  Hence 
have  log  2  =  .301  030  ;     log  3  =  .477  121  : 

log  4  =  .602  060  ;     log  3.162  277  7  —  .500  000, 

*2i 


246  LOGARITHMS.  [§  820. 

If,  indeed,  instead  of  taking  the  mean  between  10  and 
3.162  277  7,  Ave  had  taken  the  mean  between  1  and 
3.162  277  7  (that  is,  if  Ave  had  taken  the  square  root  of 
3.162  277  7),  Ave  should  have  separated  10  into  its  four  equal 
factors,  one  of  which  would  be  the  number  whose  logarithm 
is  ].  A  third  extraction  of  the  square  root  would  give  us 
one  of  its  eight  equal  factors ;  a  fourth,  one  of  its  sixteen ; 
a  fifth,  one  of  its  thirty-two  equal  factors;  and  so  on. 

Continuing  this  process,  the  twentieth  extraction  of  the 
square  root  would  separate  the  base,  10,  into  more  than  a 
3uillion  equal  factors  (1  018  57 G).  Consequently  the  log- 
arithm of  one  of  these  factors  must  be 

Todnrnr=-000  000  954. 

If  now  Ave  multiply  together  a  number  of  these  factors 
sufficient  to  produce  2,  3,  4,  &c,  and  add  together  their 
logarithms  (i.  e.  as  the  logarithms  of  the  equal  factors  are, 
of  course,  equal,  if  Ave  multiply  the  logarithm  of  one  of 
these  factors  by  the  number  of  the  factors),  the  sum  of 
these  logarithms  will  be  the  logarithm  of  the  number  pro- 
duced by  the  combination  of  the  factors.  A  combination 
of  315  545  of  these  equal  factors  will  approximately  pro- 
duce 2.     Hence  we  have 

log  2  =  315  545X-000  000  954  =  .301  030. 

§320.   Let  10*=  &  (1) 

That  is,  let  x  =  log  3. 

To  find  x,  put  equation  (1)  under  the  form, 

[l+(10-l)]"  =  [l+(3-l)]<;  (2) 

which  is  evidently  true,  whatever  be  the  value  of  t.     Then 

1+te(10-l)+<^(10-l)a+fa^^ 

(io_i  jH&c  =  i-H(«-i)4-^y(«-i)»+ 

t(U2).B~2)(d-l)"+^C'  §294" 


§321.322.]  DEVELOPMENT.  247 

Or,  canceling  1,  and  dividing  by  t, 

K1o_1)+5^=l)(1o_1).»+£fc^)(io-i)a 

+&c.  =  3-l+^(8-l)»+('^)2('~2)(3-l)3+&c. 

Developing  the  coefficients  of  10 — 1  and  3 — 1,  arrang- 
ing according  to  the  powers  of  t,  and  putting  B,  C,  &c,  B' ', 
C,  &c.  to  represent  the  coefficients  of  tl,  t2,  &c.  on  the 
two  sides,  we  have 

a:[10— 1— i(io— i)2+i(10— !)3— &C]-t-Bt-\-CT2+&c. 
—  3_  i_  i(3_  i)2_|_^(3_  x)3_ &c.-\-B't-t-Cft2+&c. 

Now  this  equation  being  true  for  all  values  of  t,  we 
have  (§  277) 

z[10— 1— 4(10— l)2+i(10— 1)3—  £(10— l)M-&e.] 

=  3-1-4(3-1)  2+i(3_l)3_JL(3-l)*+&c. 

_  3-l-4(3-l)2+A(3-l)^-&c. 

..     X  —  lOgO  —  10_1_.i(10_1)S_L.l(10_l)3_&c.-         •' 

§  321.  But  the  denominator  of  this  fraction  is  a  diverg- 
ing series ;  as  is  the  numerator,  unless  the  number  whose 
logarithm  is  sought  is  very  near  unity.  If,  however,  we 
take  the  nth  root  of  both  sides  of  ecmation  (1),  we  shall 
have 

10"  =3";     or  (10")*=  3". 

Then,  as  before, 

JL  1 

(1+10"— 1)**  =  (1+3"— 1)' ; 

and,  developing  as  before,  we  have 

S^-l-4(3~'-l)»-H(31«-l)»-&c.  u, 

x  —  log  3  =  — j- j .       ( 4 

10"— 1—4(10"—  l)2+i(10"—  l)3— &c. 

§  322.  Now  n  may  be  any  number  whatever  ;  but,  in 
order  that  the  series  may  converge  properly,  it  must  be  very 


248  LOOxARITHMS.  [§522. 

large,  and  positive.     Let  it  be  taken  so  great,  that  3"  and 

10"  may  each  be  expressed   by   1-f-  a    decimal  fraction 
whose  first  eight  places  are  cyphers.     Then  we  shall  have 

10" — 1  =  a  decimal  whose  first  eight  places  are  cyphers. 

.-.  (10" — l)2  =  a  decimal  whose  first  sixteen  places  a're  cy- 
phers. 

Hence,  the  second,  and,  with  still  more  reason,  the  sub- 
sequent terms  of  the  series  can  have  no  effect  on  the  first 
fifteen  places  of  the  denominator  (i.  e.  on  its  first  seven  sig- 
nificant  figures). 

The  first  term,  then,  will  give  the  value  of  the  denomi- 
nator correct  to  fifteen  places  of  decimals  ;  seven  of  which 
are  significant. 

The  same  reasoning  will  apply,  with  still  greater  force 
to  the  numerator ;  the  first  term  of  which  will  be  a  still 
closer  approximation  to  the  true  value  of  the  series. 

Hence,  n  being  very  large,  we  shall  have  approximately, 

i 

x(  =  \ogS)=--P~.  (5) 

10"— 1 

Let  now  n  =  2eo.     Then  we  shall  find  3",  and  10"  by 

extracting  the  square  root  of  3  and  of  10  sixty  times  in 

succession,  and  we  shall  have 

_i 

3260— 1 
x  —  lose  3  =  • 


.GO 


10-     —  1 

_  .000  000  000  000  000  000  952  894  264  074  589 
~  .000  000  000  000  000  001 997  174  208  125  505* 

x  —  log  3  =  .477  121  254  719  G62,&c 
Note.    Brigg*  took  n  =  2  54.    A  much  smaller  value  of  n  would 
give  results  sufficiently  accurate  for  all  ordinary  purposes.     The  pro- 
cess would  still,  however,  be  extremely  laborious;  and  has  been  su- 
perseded by  far  more  convenient  and  rapid  processes.     T3 u t  the  for- 


§323-325.]  FOKMtJL^E.  249 

inulte,  which  we  have  obtained,  furnish  a  convenient  method  of  ex- 
hibiting some  very  important  properties  of  logarithms. 

§  323.    For  greater  convenience  and  clearness,  we  shall 
generalize  equations  (1),  (2),  (3),  (4)  and  (5),  by  putting  y  in 
place  of  3,  and  a  in  place  of  10.     Then  we  shall  have 
ax  =  y  (G);    or  x  =  log  y  ; 

(l+a_iy<  =  (!+,,__!)<;  (7) 

,- w  rf_y-i-K.y--i)2+My-i)3-&c.  m 

j-  -  w  «  -  y.y-i-Kyy-i)2+KVy-i)3-^- . , 9 
'       °  y  ~~  v«— i— K'V«— i)2-HKV«— i)3— &c.' v 

and,  n  being  very  large,  approximately  (£  322), 

x  —  \o<*y  —  —x — .  (10) 

a"— 1 

*'  §  324.  Now  it  is  manifest,  that,  in  each  of  the  equations 
(8),  (9)  and  (10),  for  a  given  value  of  n,  [1.]  the  value  of 
the  denominator  of  the  second  member  depends  solely  on 
the  base  ;  and,  for  the  same  base  (i.  e.  in  a  given  system),  it 
remains  constant,  whatever  be  the  number  whose  logarithm 
is  required ;  [2.]  the  value  of  the  numerator  depends  sole- 
ly on  the  number  whose  logarithm  is  sought,  and  is,  there- 
fore, constant  for  the  same  number  in  all  systems  (i.  e.  what- 
ever be  the  base)  ;  and  [3.]  the  denominator  is  the  •-■ 
i'k action  of  a  as  the  numerator  is  of  y. 

§  325.  Again,  representing  the  constant  denominator  by 
f(a),  we  have  from  (8)  of  §  323 

in  which  - 

/(a)  =  or-l-i(a-l)»4i(«--l)s— i(a-l)*-Hfcc. ;  (12) 

;"^/(y)=y-i-Kz/-i)2+Ky-i)3-Kr-i)4-r^c, 


250  LOGARITHMS.  [$  326-328. 

In  like  manner,  in  another  system  whose  base  is  a',  we 
shall  have,  obviously, 

ios'y=^[y-i-My-i)2+K^-i)3-&c.]~^/y. 

Hence,      log  (y)  :  log'  (y)  =  —  :  — -..  (13) 


That 


i-:, 


§  326.  The  logarithm  of  a  number,  taken  in  different  sys- 
tems, varies  (§245)  inversely  as  the  function  of  the  base. 

a.)  If  the  base,  a,  be  given, /(a)  can  be  found  from  for- 
mula (24);  and,  on  the  other  hand,  if/(«)  be  given,  a  val- 
ue can  be  found  for  the  base,  a,  to  correspond. 

b.)  Thus,  Lord  Napier,  the  inventor  of  logarithms,  took 

f(a)  =  1,  and  constructed  the  first  table  of  logarithms  ever 

published,  on  that  hypothesis.     Consequently,  denoting  the 

Naperian  logarithm  by  Log  or  L,  and  the  logarithm  in  any 

other  system  by  log  or  I,  log'  or  V,  &c,  we  have 

Log  y  =  y_l-|(y_l)2+i(y-l)3-&c.         (14) 

.-.from  (11)  log  y  =  -=— Logy.  (15) 

J\a) 
That  is, 

\  327.  The  Nitperian  logarithm  of  any  number,  raulli- 
tiplied  by  —  -  gives  the  logarithm  of  that  number  in  the 

■J  \     J 

system  whose  base  ia  a. 

§  328.    The   quantity  — -   is   called   the  modulus0  of 

the  system  whose  base  is  a;  because  it  expresses  or  7neas- 
ures  the  ratio  of  any  logarithm  in  that  system,  to  the  Nape- 
rian logarithm  of  the  same  number. 

Hence  [§325.(13)], 

a.)  Cor.  I.  The  logarithms  of  any  number  in  different 
systems  arc  to  each  other  as  their  moduli.     And  hence, 

(c)  Lat.,  a  measure. 


<■  328.]  modulus.  251 

b.)  Cor.  ii.  The  logarithms  of  any  numbers  in  one  system 
are  to  the  logarithms  of  the  same  numbers  in  another  system 
in  a  constant  ratio,  viz.  in  the  ratio  of  their  moduli  (§  328. 
a).     Thus, 

I  2  :  I'  2  =  I  3  :  V  3  =  le  :  Ve  —  la  :  l'a=M:  M' 

[MzxA  M'  representing  the  moduli). 

»>  r      •»*■      h       l&       le    „ 

c)  Also,  since  ly  —  M.  Ly,  M=  ~  =  —  =  —  &c.  ( 1 6) 

Ly      La      Le 

That  is, 

Cor.  in.  The  modulus  of  any  system  is  equal  to  any  log- 
arithm in  that  system  divided  by  the  Naperian  logarithm  of 
the  same  number. 

d.)  Again,  since  a  is  the  base,  la=z 1. 

31=1.  „„ 

That  is, 

Cor.  iv.  The  modulus  of  any  system  is  the  reciprocal  of 
th-e  Naperian  logarithm  of  the  base  of  the  system. 

Thus,  the  modulus  of  the  common  system  is  = — -. 

J  Log  10 

Hence  (§  328) 

e.)   We  have  f(a)  =  Log  a.  (18) 

That  is, 

The  denominator  of  the  second  member  of  equation  v8) 
in  §  323  is  the  Naperian  logarithm  of  the  base. 

Note.    This  is  evident  also  from  §  324.  3,  and  §  326.  (14). 

/.)   Again,  if  e  =  the  Naperian  base,  we  have  Le  —  1. 

M=le.  (19) 

That  is, 

Cor.  v.  The  modulus  of  any  system  is  equal  to  the  log- 
arithm of  the  Naperian  base,  taken  in  that  system. 

Thus,  the  modulus  of  the  common  system  is  the  common 
logarithm  of  e. 


-'■">2  LOGARITHMS.  [§329,330. 

§329.  ^0  f(a)  —  La  —  nLan.  §316.3. 

Also  [§326.  (14)] 

Zr^=a"-l-K«"-1)2+K«;;-l)3-K«"-l)4+&c.  (20) 

11  1 

...    f(a)  [=  La  —  nLan]  z=  n[an—  1— \{an  —1)  2+&c]  ; 

=:7i(a"-l),  (21) 

when  n  is  very  large. 

/     _1_  _  J.  _ L    ^  — 1 

MXf{a)  ~~  La~~  nL{n+/a) )  ~n("\/a—l)  ' 

=i.^-.  (22) 

n    — 
an—\ 

And,  for  the  common  system,  taking  n  =  260,  we  have 
approximately  (21) 

/(a)  =  Zl0  =  ??Z10;r=2Go.(i026O_l). 
1         _1_    _L %   _1_  1 


£10~2«0*      -L-  2co-2co 

•JQO6  0 i  */ IV 1 

Now  we  have 

—  =  0.000  000  000  000  000  000  867  361  737  988  &c.,  and 

2 |;  ° 

j 

1026O=  1.000  000  000  000  000  001  997  174  208  125  &c 

_  867  361  737  988  &c. 
—  1  997  174  208  125  &c." 

M{—  S-  =  h)  —  0.434  294  481  903  251  &c.  (2-3 

§  330.  k.)   If  we  take /(a)  =  1,  then  (§§  326.  b;   328./) 

a  —  e,  and  [§  329.  (21)],  approximately, 

i  11  11 

\  =in(e"  —  l);        or-  =  en—  1;     orl-f-  =  c. 

1\" 


(1+-).  .-.(W4J 


§  331,  332.]      NAPERIAN  BASE  AND  MODULUS.  253 

.  =  1+B*  +  »fcL>(i)  Vi!!=i}(^!)(I)3+&„, 

1    n  1.2     \w/  1 .2  .d        w 

or,  reducing,  and  neglecting  all  the  terms,  which  have  n, 
n-,  &c.  in  the  denominator,  inasmuch  as  (n  being  =  260) 
they  cannot  affect  the  first  sixteen  places  of  decimals, 

e=1+i+iVi^s+nnrT+&<!'  <-" 

e  =  2.718  281 828  459  045  &c.  (25) 

Note.    To  find  the  sum  of  any  number  of  terms  of  this  3eries, 
divide  the  second  term  by  2  for  the  third ;  the  third  term  by  3  for  the 
fourth,  and  so  on;  and  then  add  the  terms.     Thus, 
1 

2)  1 

3)  0.5 

4)  0.1  666  666  666  666  666  666  666  666  666  666  666  &c. 

&c.   0.0  416  666  666  666  666  666  666  666  666  666  666  &c. 

§  331.  i.)  If  we  take        f(a)  =1,  §  330. 

we  have  m(=~  =  1-  =  1)  =  1.  [(26) 

V    /(«)      f(e)      ^e' 

That  is, 

The  modulus  of  (he  Naperian  system  is  unity. 

This  is  evident  also  from  §§  327,  328,  c,  d,  e,  or/. 

Notes.  (1).  On  account  of  this  property  the  Jifoperian  system 
is  sometimes-called  the  natural  system;  it  being  the  standard  to 
which  all  other  systems  are  referred  by  their  moduli.  (2.)  The  Na- 
perian  system  is  also,  in  general,  most  convenient  for  algebraic 
(§315.  g)  investigations  and  expressions;  because  the  modulus,  be- 
ing unity,  need  not  be  written.  (3.)  Naperian  logarithms  are  also 
sometimes  called  hyperbolic  logarithms,  because  they  express  cer- 
tain areas  connected  with  the  hyperbola.  This  name,  however,  is 
less  appropriate,  because  other  systems  of  logarithms  have  the  same 
property  with  reference  to  different  forms  of  the  hyperbola. 

§  332.  We  have  [§§  328  ;  325.  (12)] 
1  1 


M: 


'/(a)      a-l-Ka-l)2+i(a_1)3_&c/ 

ALG.  22 


~54  LOGARITHMS.  [§333,334. 

.-.,§323.(8),       log  y  =  M[y— I— l{y—\y +&<>.■].    p,7) 
I  i  i 

log  y»=M[y»—l—ls(y»—iyjt&Q.] ;  (28) 

1  j_ 

.i  logy»  =  M(yn—l),  (29) 

when  n  is  very  large. 

Also  (§327)        logy  =  MIuogy.  (30) 

-      i 
§  333.    Let  anl  =  yn,  n  being  a  very  large  positive  num- 
ber, as  2eo,  and  y  being  a  small  number  greater  than  uni- 

ty ;  so  that  yn  may  differ  but  very  little  from  tm%,  and  -, 

n 
from  aero.     Then  [§§  312  ;  332.  (30),  (29)] 

x  I  11 

-  =  \og  y»  =  M.  lag  y*  =  M(yn— I). 

fit 

x'         1  I  re"  1  i 

So     -  =  ly'»  =  M{y<»  — 1)  ;    —z=  ly"n  —  M(y">j:—l)  ;  &c. 
n  n 

1111  I  1       - 

lyn  :ly'n  :  ly"n  —  yn—l  :  y'n—\  :  y"n—l. 

l         l 

»r  log  yn=yn—  1.  §245. 

That  is,  approximately, 

1.  The  logarithms  of  numbers  very  wear  ttm'Jy  are  to 
:jach  other  as  the  differences  of  the  numbers  from  unity. 

Thus,        log  1.000  001  =  .000  000  434  294 ; 

log  1.000  002  =  .000  000  868  588 ;  and 

.000  000  434  294  :  .000  000  868  588  =.000  001  :  .000  002. 

Also  (§238) 

ii.  1         JL         1       1         1       1 

lyin—ly*  :  ly"n—lyn=y,n—yn  :  yH«—y\ 

That  is,  approximately, 

2.  The  differences  of  the  logarithms  of  numbers  very  near 
unity  are  to  each  other  as  the  differences  of  the  numbers. 

See  example  under  1,  above;  and  under  §  335.  b,  c. 

§  334.  a.)  Let  y  be  not  <2,  and  w  =  2G0,  so  that  y"may 


§  335.]  DIFFERENCES  OF  LOGARITHMS.  253 

be  a  very  large  number ;  also  let  D  =  a  small  number,  as 
1,  2,  &c.     Then  (§§  31G.  2 ;  115  ;  333.  1) 

ii'  yn  is  constant  (5  247.  3).     That  is,  approximately, 

If  large  numbers  differ  by  quantities  very  small  in  com- 
parison with  themselves,  the  differences  of  their  logarithms 
will  be  as  the  differences  of  the  members. 

Thus,  in  the  common  system,  if  the  logarithms  are  car- 
ried to  only  seven  places  of  decimals,  we  have 

n0000  =  4;      1 10  001  =  4.000  043  4; 
HO  002  =  4.000  080  8  &c. ; 

where,  for  equal  increments  of  the  number,  we  have  equal 
increments  of  the  logarithm. 

Note.  We  must  not  extend  the  series  far,  because  the  difference? 
of  the  numbers  would  cease  to  be  very  small  compared  with  the  num- 
bers themselves. 

t 

§  335.  b.)   We  have,  by  (26)  and  (29), 

11  11 

Lyn  =  yn—1,  and  Lyln  =  yi"—\. 

1  III 

Ly'n—Lyn  =  y"l—y>\ 

That  is,  approximately, 

The  difference  of  the  Naperian  logarithms  of  two  num- 
bers very  near  unity  is  equal  to  the  difference  of  the  num- 
bers. 

Thus,  Log  1  =0; 

Log  1.000  001  =  0.000  000  999  999,  or  0.000  001 ; 

Log  1.000  002  =  0.000  001  999  998,  or  0.000  002. 

That  is,  the  numbers  differing  by  .000  001,  their  Nape- 

s-ian  logarithms  differ,  within  an  extremely  small  fraction. 

by  the  same  quantity. 

c.)  In  any  system  whatever,  we  have 

I  I  i  i        '  . 

log  y»  =  M(y»—1 ) ;     log  yln  —  M{xjn—\ ). 


256  LOGARITHMS.  [§336, 

I  1  1    i 

log  y> "-log  y«  =  M(y>»-f). 

That  is,  in  any  system,  approximately, 

The  difference  of  the  logarithms  of  two  numbers  very 
near  unity  is  equal  to  the  difference  of  the  numbers  multi- 
plied by  the  modulus  of  the  system. 
Thus,  in  the  common  system, 
log  1  =0; 

log  1.000  001  =  0.000  000  434  29  ; 
log  1.000  002  =  0.000  000  868  58. 
That  is,  having  cliff,  numbers  =  .000  001, 
we  have   diff.  logarithms  =  .000  OOlX-434  29.  §  329.  (23) 

Note.    If  we  make  M=l,  the  logarithms  become  Naperian, 
and  this  principle  becomes  identical  with  the  preceding. 

d.)    Reasoning  as   in  §  334,  we  shall  find,  that,  if  any 
number  whatever  receive  an  increment  very  small  in  com- 
parison with  itself,  the  corresponding  increment  of  the  log- 
arithm is  approximately  ecptal  to  the  modulus  into  the  fo- 
ment of  the  number,  divided  by  the  number.     Thus, 

l(ynJrD)-ly"  =  l(lJr^;)=M^.    [See  §§337.  a; 

332.  (28),  (29)].        Thus  (§334), 

log  10  001— log  10  000  =  .434X  ro  J  o  o  =  -000  043  4' 

So,     log  9865  =  3.994  097,   log  9806  =  3.994141.     And 

log  9866— log  9865  =.434  294x^5  =  -000  °44- 

§336.  e.)  If  a  number  exceed  unity  by  a  very  small 
quantity,  its  logarithm  exceeds  zero  by  a  very  small  quan- 

tity.     Now,  §332.  (29),  M=  -^pU 

Therefore,  the  modulus  approximately  expresses  or  mea- 
sures the  ratio  (§  230)  of  the  infinitesimal  excess  of  the 
logarithm  above  zero,  to  the  corresponding  infinitesimal  ex- 
cess of  the  number  above  unity, 


§  337.]  COMPUTATION.  257 

m,     .000  000  434  29        .6jor 
Thus  (§  335.  b),        M=  -^qoqqoI —  =  *434  29* 

/.)  Or  (§335.  d),         M=       .X_/  ,    '  •       Thatis, 

The  modulus  approximately  expresses  the  ratio  of  an 
Infinitesimal  increment  of  any  logarithm,  to  the  correspond- 
inc  increment  of  the  corresponding  number,  divided  by  tli> 
number  itself. 

Thus  (§  335.  d)      M=  '00°1044  =  .434  294. 

§  337.  Putting,  in  (27)  of  §  332,  l+y  instead  of  y,  and 
of  course  y  in  place  of  y — 1,  we  have 

log(i+y)  =  ^(y-^2'+^3-|2/4  +  &c.).     («) 

Putting  — y  for  y  in  («),  we  have 

l0g(l-y)  =  J/(-y-I2/2_^3_^4_&c.)>        (/)) 

Subtracting  (5)  from  (a), 

1-4-w  E 

«(l-^)-'(l-jr)='^=2^(jr+i»?-+4e.)-    W 

•••     l0g(l+y)=l0g(l-3/)+2iJ/^+^3+i3/5H_&c.).    (31) 

This  series  converges  with  tolerable  rapidity,  when  y  is 
a  small  fraction.     Thus  if  y  =  .1,  we  have 
.      1-f.l  11 

log  n-iog  9  =  2ir(i5+g(L)-  +^-+&c). 

,       log  11  =  log  9+2^(^  +  3^-  +^-5+&e.). 

Here,  if  no  more  than  seven  places  of  decimals  ai*e  re- 
quired, the  fourth  term  of  the  series  may  be  neglected. 

Now,  §  329.  (23),         M—  0.434  294  48 ; 

and,  §§  316.  3  ;  322,     log  9  =  2  log  3  =  .954  242  5. 

*22 


258 


LOGARITHMS. 


[§338,  339. 

log  11  =  .954  242  5-J-.868  588  96(.l+.000  333  3+&c). 
log  11  =  1.041392  8. 


log 


§  338.  Making  in  (c) 

1  1+y      *4-l 


and 


=  "*^i+SKKF+ 


H   '  8(2*fl)«   '  5(22+1)5  +&c-)' 
log  (=+1)  =  log  HJJr^  +  p^s  +&c.).(32) 
Thus,  if  *  =  10,  we  have  z-\-l  =  11  ;  and 

log  11  =  log  10+.868  588  96(^-  +  -^-3  +&c.). 
This  series  may  be  summed  thus ; 

2*4-1=    21.868  588  96       =23f 
(2*-f-l)2  =  441 
(2*4-1)  2  =  441 


041  361  38-r-l  --.041  361  38 
93  794-3  =  31  26 

214-5  =  4 


.041  392  68. 


log  11  =  log  10  4-  .041  392  68  =  1.041  392  68. 

This  series  converges  much  more  rapidly  than  the  preceding. 
Many  still  more  rapidly  converging  series  have  been  devised.  We 
shall  give,  however,  but  a  single  example. 


u 


!— 1,  and  (32)  be- 


§339.   Let*4-l  =  ?o2.     Then  ; 
comes 

log  ..  =l„g  («._l)+2Jf  (s-^.  +  _  —+&*■)■ 

Or,  as  log  w2  =  2  log  u, 

and  log  («- — 1)  =  log  («4-l)4~log  (w— 1), 

we  have        log  (?t  -j- 1)  =  2  log  u  —  log  («  —  1)  — 

1 


Uf(i5r=r+ 


3(2«2  — l)3   '  5(2u2— 1) 


1_Ti^+&4(38) 


§  340.]  EXPONENTIAL  THEOREM.  259 

Thus,  if  u  =  12,  we  have 
log  13  =  21og  12-Iog  n-2^(^  +  374?+&c.). 

Now  2  log  12  =  2  (log  34-log  4)  =  2  (log  3+2  log  2), 
=  2(.477  121  25+.602  060)  =  2.158  364  ; 
and  (§  336)  log  11  =  1.041  392  68. 

The  series  may  be  summed  thus  ; 

2«2— 1  =  287  .868  088  96       =  2,1/ 
(2**2— l)2  =  62  369 


.003  026  44-KL  =  .003  066  44 
5-^3  =  2 


.003  026  45. 


log  13  =  2.158  362  5—1.041  392  68— .003  026  45. 
=  1.113  943. 
For  all  lai-ger  numbers,  the  first  term  of  this  series  will 
give  the  value  correctly  to  eight  places. 

Note.    The  logarithms  of  the  prime  numbers  only  need  be  com- 
puted by  such  processes ;  the  logarithms  of  all  other  numbers  being 
found  by  the  proper  combination  of  the  logarithms  of  primes.     Thus, 
log  4  =  2  log  2 ;     log  6  =  log  2+log  3 ;  &c. 
The  logarithms  of  2  and  3  may  be  found  from  formula  (32),  and 
the  logarithms  of  5  and  7  from  (32)  or  (33). 

EXPONENTIAL  THEOREM. 

§  340.  In  the  equation  ax  =  y,  we  have  found  x  in  terms 
of  y ;  i.  e.  a  logarithm  in  terms  of  the  corresponding  num- 
ber. 

We  shall  now  find  y  in  terms  of  x ;  i.  e.  a  number  in 
terms  of  its  logarithm. 

Put        y[=  a*  =  (l+o-l)*]  =  [(l+«-l )»]""; 

n  being  any  number  whatever,  and  of  which  the  value  of  y 

is  entirely  independent  (§  276  ;  277  ;  280.  e). 

Developing, 
(1+^l)»  =  l+n(0_l)_L.^d)(a_l)2+&c.. 


260  •     LOGARITHMS.  [§34l, 

or,  (1+a—iy  =  1+An+B?i*+Cn3+  &c. ; 

A,  B,  C,  &c.  being  functions  of  a  ;  and,  evidently, 

A  =  a— 1— J-(«— 1)2-B(«— 1)5~ &c-  =  La.    §  326.  £. 
Then  we  have 

y = [(i+o-i)*]» ;=  [i+(^+^2-Hbc.)]s: 

...  y  =  l+^(An+B^+&c.)+  ^=J} (^n+i^+&c.) 2 
r^-rc) (*-2»)_ ( j w+jR|fl+&c ,}  3+&c. .  $  295.  *. 

or  ?/  =  l+a:(J+JBH-&c.)+  ^^\-4+JBh+&c.)  2 

f    x(x — n)(x — 2n) 


1.2.3 


-(^4+^+&c.)3+c&c. 


A2x2       A%xz 
■.(§277)    #  =  «'=l+^+— +— ~3+&c.     (34) 

,   (Za)2a:2   ,   (Za)^3    ,    0 
Or     af=l+Za.aH-iY72~+ii-^73-+&c      (3o) 

§  341.  a.)  We  know  the  value  of  A  from  §§  325.  (12) ; 
328.  e.  But,  if  we  did  not  know  it,  we  might  find  it  from 
the  equation  (34)  itself.     Thus, 

Let  x  =  -j.  Then    (§330) 

^=1+I+l^  +  i4.-3  +  IT2L3-.T+&C-  =  e-      <36) 
.-.  (§52.  N.)  a  —  eA.  (37) 

That  is,  -j-  is  the  logarithm  of  e  to  the  base  a  (§  328.  d, 

/') ;  and  A  is  the  logarithm  of  a  to  the  base  e  (i.  e.  the  Na- 
perian  logarithm  of  a  [§  328.  e]). 

5.)  Or  thus  ;  taking  the  logarithms  of  both  members  of 

(37), 

log  a  =  A  log  e.  §316.3. 


§342-344.]  EXPONENTIAL  EQUATION.  201 

log  a  1  1        ,    . 

A  =  r-2 —  =  Lo°-  a  =  -. =  ^rp,  a  being  the  base. 

log  e  log  e      M 

That  is,  A  is  the  reciprocal  of  the  modulus  of  the  sys- 
tem of  logarithms  whose  base  is  a. 

Note.  The  logarithms  may  be  taken  in  any  system,  provided 
both  be  taken  in  the  same;  the  logarithms  of  two  numbers  having  the 
same  ratio  in  one  system  as  in  another  (§  328.  b). 

§342.  c.)  We  have  found  (§341)  the  value  of  A,  in 
terms  of  a.  We  may,  if  Ave  prefer,  assign  a  value  to  A, 
and  find  the  corresponding  value  of  a. 

Thus,  let  A  =  l. 

Then,  «'  =  1+*+  ~  +  —^  +&c. 

Making  x  =  1 ,       a  —  1+1+  —%  +  OT3+&C" =  e' 

§  343.  d.)  This  is  called  the  exponential  theorem ;  and 
y,  in  the  equation,  y  =  ax,  is  called  an  exponential  func- 
tion of  x.  On  the  other  hand,  x  is  called  a  logarithmic 
function  ofy ;    being  the  logarithm  of^  to  the  base  a. 

e.)  These  two  classes  of  functions  ai*e  also  called  teans- 
cendentai/  functions ;  as  transcending  the  elementary  op- 
erations of  Algebra. 


EXPONENTIAL  EQUATION. 

§  344.   The  equation,         ax  =  b, 
is  called  an  exponential  equation.     If  a  is  the  base  of  a 
system  of  logarithms,  we  have  simply 

x  —  log  b. 

But  if  a  is  not  the  base  of  a  system  of  logarithms,  take 
the  logarithms  of  both  sides  of  the  equation,  in  any  system. 
The  common  system  is  usually  most  convenient.     Then, 
log  (ar)  =  log  b  ;       ov  x  log  a  =  log  b.         §  316.  3. 

(d)  Lat.  transcendo,  to  exceed,  surpass. 


262,  LOGARITHMS.  [§  345 

_log  b 
log  « 

1.  Given  12*  =  20,  to  find  x. 

x  log  12  =  log  20 ;  i.  e.  1.079  181  25a:  =  1.301 030. 

log  20      1.301030 

x  =  f^T^  =  i  n~oiQio*  =  L205  57 &c- 
log  12      1.0/9  181  25 

2.  Given  603'  =  7,  to  find  x.  Am.  x  =  .475  273. 

3.  Given  125x=  25,  to  find  x.  Am.  x  =  §. 

§  345.  c?.)  If  the  equation  be  of  the  form, 

xx=  b, 
we  have  x  log  a;  ==  log  b. 

This  equation,  can  be  most  conveniently  solved  by  the 
method  of  trial.  For  this  purpose,  find  by  trial  two  ap- 
proximate values  of  x.  Substitute  these  values  successive- 
ly in  the  equation, 

x  log  x  =.  log  b, 
and  note  the  error  in  each  result.     Then 

Diff.  of  the  results  :  Diff.  of  the  assumed  numbers  =  the 
•  rror  in  either  result :  the  correction  to  be  applied  to  the  cor- 
responding assumed  number. 

This  correction,  being  applied,  will  give  a  nearer  ap- 
proximation to  the  true  value  of  x.  This  new  value  may 
now  be  taken  as  one  of  the  assumed  numbers,  and  a  still 
closer  approximation  obtained ;  and  so  on. 

1 .    Given  x':  =z  100,  to  find  x. 
Here  we  have        x  log  x  '==  log  100  =  2. 
Also,  since         33  =  27,  and  44  z=.  25  G,    the  value  of  x 
lies  between  3  and  4. 

Substituting  3  and  4  successively,  we  have 

3  log  3  =  3  X  0.477  121  25  =  1.431  363  75  ; 
whence        2  — 1.431  363  75  =  .568  636  25,  error; 


§  34:5.]  INTEREST. — POPULATION.  263 

also  4  log  4  =  4  x  0.602  059  99  ==  2.408  239  96 ; 

whence         2  —  2.408  239  96  =  —  .408  239  96,  error; 
and  0.976  876  21  =  difference  of  results. 

.967  876  :  1  =n—  .408  239  96  :  —.418,  correction. 
4  —  .418  =  3.582  =  x  nearly. 

Again,  we  find  x>  3.582,  and  <3.6.  Therefore,  substi- 
tute these  values,  and  repeat  the  operation.     Thus, 

3.582  log  3.582  =  3.582  X  0.554 126  5  =  1.984  881. 
2  — 1.984  881  =  .015  119,  error. 

3.6  log  3.6  =  3.6  X  0.556  301  9  =  2.002  689. 
2  —  2.002  689  =  —  .002  689,  error. 
Also,  .017  811  =  difference  of  results. 

.-..      017  811  :  .018=— .002  689  :  —.002  717,  correction, 
3.6  —  0.002  717  =  3.597  283  =  x  nearly. 
That  is,  3.597  2833-597283  — ioo. 

2.  Given  x' '—  5,  to  find  x.  Ans.  x  =  2.1293. 

3.  Given  x*  =  2000,  to  find  x. 

Ans.  x  —  4.827  822  63. 

4.  Given  a,  m  and  I  of  an  equimultiple  series,  to  find  n 

<^64-J)-  A,B=!2|lri21?+1. 

I02;  m 

5.  Let  a  =■  2,  I  =  162,  and  m  =  3  ;  and  find  n. 

Ans.  n  —  5. 
6s   In  how  many  years  will  p  dollars  amount  to  A  dol- 
lars, at  r  per  cent,  compound  interest  ? 

We  have  (§  258.  5)  A  —p{l-{-r)t. 

.  low  A — log  p 

Ans.t—    °     n,\J. 
log  (1+r) 

7.  In  what  time  will  $100  amount  to  8200  (i.  e.  in  what 
time  will  a  sum  of  money  double  itself  )$  at  6  per  cent 
compound  interest?  Ans.  11.89  years 

Here  p  =  100,  A  =  200,  and  1+r  =  1.06. 


204  LOGARITHMS.  [§845. 

Notes.  (1.)  The  solution  of  most  questions  relating  to  com- 
pound interest  may  be  greatly  facilitated  by  the  use  of  logarithms. 
(2.)  The  formula?  of  compound  interest  apply  also  to  the  increase  of 
population  in  a  country. 

8.  Find  r  fro'm  the  formula,  A  =p(l-\-ry.     See  §  258. 

5?  9-  .       ,       ...  ,    x       log  A — loo;  p 

Ans.  log  (1+r)  ==  — 5Ll . 

9.  The  population  of  the  United  States  in  1830  was 
12  866  000,  and  in  1840, 17  068  000.  What  was  the  year- 
ly rate  of  increase  ? 

Here  A  =  ll  068  000,  p '=.  12  866  000,  and  t  — 10. 

.      „  ,    .      log  17  068  000— log  12  866  000 
.-.       lug(l+r)=-2 , 


10 

7.232183—7.110118 


=  .012  206  5. 


10 
1  -f  r  =  1.0285  ;  and  r  =  .0285  =  2J$  per  cent. 

10.  At  the  same  rate,  what  will  be  the  population  in 
1850? 

Here p  = 17 068  000;  r=.0285,  and  <  =  10. 
A[=p{l-\-ry~]  =  17  068  000  (1.0285) }  °. 
log  A  =  log  17  068  000+10  log  1.0285. 

Ans.  A  =  22  654  000. 

11.  In  how  many  years  will  the  population  amount  to 
50  000  000  ?  Ans.  In  38.24  years  from  1830. 

12.  If  the  number  of  slaves  in  the  United  States  in 
1830  was  2  009  000,  and  in  1840,  2  487  000,  what  was  the 
yearly  rate  of  increase?  Ans.  2%  per  cent. 

13.  At  the  same  rate,  what  will  be  the  number  in  1850  ? 
in  1860  ?   Ans.  3  078  700,  in  1850  ;  3  811  000,  in  1860. 

14.  The  population  of  Virginia  in  1830  was  1211  400. 
and  in  1840,  1  239  700 ;  that  of  New  York  in  1830  was 
1  918  600,  and  in  1840,  2  428  900.  What  was  the  yearly 
rate  of  increase  in  each  state  ? 

Ans.  In  Virginia,  .0023,  or  less  than  \  of  1  per  cent; 
in  New  York,  .0238,  or  more  than  2^  per  cent. 


CHAPTER  XVI. 


THEORY  OF  EQUATIONS. 


§  346.  We  shall  confine  ourselves  here  to  the  considera- 
tion of  equations  containing  but  one  unknown  quantity. 

1.  If  the  exponents  of  the  unknown  quantity  in  such  an 
equation  be  all  integral,  or  if  their  differences  be  all  inte- 
gral, the  degree  of  the  equation  is  correctly  expressed  by  the 
difference  beticeen  the  greatest  and  the  least  of  those  expon- 
ents (§§40.  a;51.b). 

2.  But,  if  the  difference  between  any  two  of  the  expon- 
ents be  fractional,  this  difference  between  the  greatest  and 
least,  obviously,  may  not  express  the  degree  of  the  equa- 
tion.    Thus,  evidently, 

x 
x'2  -J-  ax  -\-  bx^  -\-  c  —  0 

is  not  of  the  second  degree. 

Reducing,  however,  the  exponents  to  a  common  denom- 
inator, we  have 

e  3  i 

x^  -\-  ax3  -f-  foe3  ~f-  c  =  0, 

which  may  be  said  to  be  of  the  §  degree.     In. fact,  if  we 
make  3*/x  —  y,  we  shall  have 

y6  +  «y  3+  5y + c  =  0  j 

an  equation  of  the  sixth  degree  in  respect  to  y  (i.  e.  in  res- 

i 
pect  to  x3). 

Hence,  when  the  difference  between  any  two  of  the  ex- 

ALG.  23 


ii66  THEORY  OF  EQUATIONS.  [§  347,  348. 

juts  is  fractional,  the  degree  of  the  equation  is  the  dif- 
ference between  the  greatest  and  least  exponents,  expressed 
in  terms  of  the  least  common  denominator  of  all  the  expon- 
mts.     Thus, 

1-  3  2 

.>  -  -\- ax'J  -\- b  (=:  xG  -\-  ax5  -\-  b)  =  0  is  of  the  §  degree. 

i  i  i 

x3  -f-  ax1  -4-  bx°  -J-  c  =  0  is  of  the  f%  degree. 

347.  It  is  evident  that  equations  of  this  kind  can  be 
expressed  in  integral  degrees,  by  reducing  their  exponents 
to  a  common  denominator,  m,  and  substituting  a  new  un- 
known quantity  for  the   mth   root   of  x  (i.  e.  by  putting 

y=.y). 

Hence,  we  shall  need  to  consider  equations  of  integral 
legrees  only,  and  shall  suppose  them  reduced  to  the  fol- 
lowing form,  viz. 

xn  +  A1xn-3l^-Azxn~s  .  .  .  -f  A^^+Al—O.    (1) 

We  shall  also  assume,  that  every  equation  has  at  least 
>ne  root. 
Note.    A  single  symbol,  as  X,  or  f{x),  is  sometimes  put  for  the 
il  member  of  an  equation.     Thus,  X=0,  or /(a:)  =0. 

DIVISIBILITY. — ROOTS. 

18.  Let  a  be  a  root  (§  39)  of  equation  (1).     Then, 

J-i^/'-'+ioa"-2  •  •  •  .+An.1a-\-An  —  0. 
.-.      A,r=~an  —  Alan-1—A2an-2  ....  —An.xa. 

Substituting  this  value  of  An  in  (1),  we  have 

.--—  an)-\-Al(x,'~l  —  an~l)   .  .  .  +AH-1(x  —  a)  =  0. 

Now  this  expression   is   divisible  by  x— a  (§§  81,  9G). 
Hence  (compare  §  213.  5), 

If  a  be  a  root  of  the  equation, 

rJ-i/"1 i-A^vc  +  An  =  0, 

Hie  first  member  of  the  equation  is  divisible  by  x—a. 


$348.]  DIVISIBILITY. — ROOTS.  '-''''. 

a.)  This  principle  may  be  demonstrated  otherwise  ;  thus. 

If  v»Te  actually  divide  the  first  member  of  (1)  by  x—a.  we 
shall  have,  representing  the  quotient  by  Q,  and  the  re- 
mainder by  i?, 

f(x)  =  x"  -f- A xx"~ i  .  .  .  +  A  ,  =  (x  —  a><?  +  #  =  0.    : 

Now,  if  «  is  a  root   of  the  equation,  the  supposition. 
,r  —  a  (i.  e.  x — a  =s  0), 
reduces  the  first  member  of  (1)  to  zero  (§§  39,  211). 

H  =  0,  and  £/ze  division  is  perfect  (§  82.  </ . 

Thus,  4,  5  and  —  1  are  roots  of  the  equation, 
a;3— ^  8*2^11*4- 20  =  0. 

See  if  the  first  member  is  divisible  by  x  —  4,  x  —  5,  and 

b.)  If  «  is  not  a  root  of  the  equation,  the  substitution  of 
a.  for  x  will  not  reduce  the  first  member  of  (1)  to  zero.  In 
that  case,  we  shall  have,  from  (2), 

/(a)  =  an  +  J1a"-i   .  .  .  -^An^1a-\-4n  =  ^    That  is. 

If  a  polynomial,  a /miction  of x,  of  the  form, 

xn-JrAlx"~l-{-A2x,1-:i +.4n, 

he  divided  by  x — a,  the  remainder  will  be  the  same  fund > 
of  a,  that  the  given  polynomial  is  of  x  ;  i.  e.  it  will  be  what 
the  given  polynomial  becomes,  when  a  is  substituted  for  x. 
■See  §211.  1. 

Notes.  (1.)  The  remainder  is  independent  of  or.  For,  if  it  con- 
tained x,  the  division  might  be  continued  farther.  R,  therefore, 
since  it  does  not  contain  x,  will  have  the  same  relation  to  a,  whatever 
value  is  given  to  x.  (2.)  It  is  evident  also  from  this  principle,  that. 
if  a  is  a  root  of  the  equation,  the  remainder  will  be  zero,  and  the  di- 
vision perfect. 

1.  Divide  x3  -}-^41x2-|-  A2x-]-A3  by  x  —  a. 

Bern.  a$  -j-  A  xa}  -f-  Aaq + A9> 

2,  Divide  x?  —  8x2  +  1  Ix  +  20  by  x  —  a. 

Rem.  a3_^8a2-f-llrt-f-2M, 


268  THEORY  OP  EQUATIONS.  [j  349,  85& 

§  349.  c.)  Conversely,  if  the  first  member  of  (I)  be  di- 
visible by  x — a,  then 

7?  =  a"  +  ^1an-i  +  J2an~2 +Jn  =  0; 

i.  e.  the  substitution  of  a  for  x  satisfies  the  equation  (§  39) ; 
and,  therefore,  a  is  a  root  of  the  equation. 

d.)  Hence,  to  determine  whether  a  is  a  root  of  the  equa- 
tion, x"  +  Axxn-i -\-An=:(), 

we  have  only  to  divide  the  first  member  by  x — a.     And, 

(1.)  If  the  division  is  perfect  (§  82.  g),  a  is  a  root;  (2.) 
if  it  is  not  perfect,  the  remainder  is  the  value  of  the  first 
member,  with  a  substituted  for  x  (§  348.  b). 

§  350.    1.   Find  whether  3  is  a  root  of  the  equation. 
a;»  -a;*  —  25a;3  -4-  85a:2  —  96a;  +  36=0, 
Divide  by  x  —  3 ;  thus  (§  86), 


1  _  i  __  25  +  85  —  96  +  36 
+  3  +    6  —  57  +  84  —  36 


1 

+  3 


1  +  2  —  19  +  28  —  12,      0,  the  remainder, 
Hence,  the  remainder  being  zero,  3  is  a  root. 

2.  Find  whether  4  is  a  root  of  the  same  equation. 

In  performing  these  divisions,  the  first  coefficient,  being 
1,  need  not  be  written.     Thus, 

1  —  1  —  25  +  85—    96  +   36     4 

+  4  +  12  —  52  +  132  +  144 
1  +  3  —  13  +  33+    36,  +  180,  the  remainder. 

Consequently,  4  is  not  a  root ;  and  the  substitution  of  4 
for  x  reduces  the  first  member  to  180. 

3,  What  does  the  first  member  of  the  equation, 

x*  _  7^3  _  20a:2  +  30a;  —  48  r=  0, 
become,  when  7  is  substituted  for  x  ?  Ans.  —  818. 

This  may,  of  course,  be  determined  by  the  actual  substitution  of  7 
for  x.  But  we  arrive  at  the  same  result  much  more  conveniently  bj 
dividing  by  x —  7S  as  in  the  preceding  examples. 


5  351.]  NUMBER  OF  ROOTS. 

4.    What  does  the  first  member  of  the  equation, 

a*  — 20x-T-96  =  0, 

become,  when  7  and  9  are  successively  substituted  for  .<■  r 

Ans.  5,  and  — 3. 

NUMBER  OF  ROOTS. 

§  351.  Let  ax  be  a  root  of  the  equation, 

X  =  x*+  Axxn~ 1  +  &c.  =  0.  1 

Then  if  we  divide  by  x  —  av  we  shall  have,  evidently, 

an  equation  which  will  be  satisfied,  if  either  of  its  factors 
be  equal  to  zero.     Making 

x"~l  +  B1xn~3  +  Sep.  =  0, 

and  supposing  a2  to  be  one  of  its  roots,  the  primitive  equa- 
tion will  take  the  form  (§  348), 

X—  (x  —  ax)(x  —  a2)(x"~2-{-  C^~-  .  . «.  +  C„_2)  =  <*. 

AVe  may,  obviously,  proceed  in  this  way,  diminishing  the 
degree  of  the  polynomial  by  unity  at  each  division,  till  w 
have  taken  out  n  factors  of  the  form  x  —  a. 

X=xn  +  A1xn-*  ....  -\-A.n_lX-\-Au  = 

(x  —  al)(x  —  a2)(x  —  a3) (x  —  an)==0.     (2) 

Now  this  equation  will  be  satisfied,  if  any  one  of  its  n 
factors  be  equal  to  zero ;  i.  e.  if  x  be  equal  to  any  one  of 
the  n  quantities,  «x,  a2  .  .  aa.     Therefore, 

1.    Every  equation  of  the  form, 

X=xn  +  A1x''~i +A  =  0, 

can  be  resolved  into  n  binomial  factors,  of  the  form,  x— 

(2.)  Every  equation  has  as  many  roots  as  there  are  mite 
in  its  degree.     See  §  213.  1,  2. 

Thus  (§  348.  a),  the  equation, 

£3  _8x2  -|- 1  \x  +20  =  (x  —  i)(x  —  5)  (x  + 1)  -  0, 
has  the  three  roots,  4,  5,  — 1. 

*23 


270  THEORY  OF  EQUATIONS.  [§352,353. 

§  352.  Suppose  b,  a  quantity  different  from  any  of  the 
roots  a1,  a2,  a3,  &c,  to  be  a  root  of  equation  (2).  Then 
we  have 

(5.—  ai)(b  —  a2)(b  —  a3)  .  .  (b  —  an)  =  0, 

an  evident  absurdity ;  because,  by  hypothesis,  b  being  not 
equal  to  any  of  the  quantities,  ax,  a2,  &c.,  no  one  of  the 
factors,  b — av  b — a2,  &c.  can  be  equal  to  zero.     Hence, 

The  number  of  roots  of  an  equation  cannot  be  greater 
than  the  number  of  units  in  its  degree. 
Hence  (§§  351,  352) 

§  053.    The  number  of  roots  of  an  equation  is  al- 
ways equal  to  the  number  of  units  in  Us  degree. 

a.)    These  roots  may  be  all  real ;  or  part  or  all  of  them 
may  be  imaginary  (§  216). 

b.)  Again,  they  are  not  always  different  from  one  anoth- 
er.    Any  part,  or  all  of  them  may  be  equal  (§  205). 

An  equation  will,  of  course,  contain  equal  roots,  when  it-: 
first  member  contains  equal  factors. 

Thus,  the  equation, 

x3  — 3x2  +  3x  — 1  =  {x  —  l)(x  —  l)(x  —  1)  =  0, 
has  three  roots,  each  equal  to  1. 

c.)  If  we  know  a  part  of  the  roots  of  an  equation,  wc 
may  find,  by  dividing  by  the  corresponding  factors,  the 
equation  of  a  lower  degree,  which  contains  the  remaining 
roots  (§351). 

1.  One  root  of  the  equation, 

x*  —  9x3  -f  19x2  +  9x  —  20  =  0, 
is  1.     Find  the    equation  which   contains  the  remaining 
roots.  Ans.  x3  —  8x2  _j_  \\x+  20  =  0. 

*   *  $ 

2.  Another  root  of  the  same  equation  is  4.     Find  the 

equation  containing  the  other  two  roots. 

Ans.  x2  —  4x  —  5  =  0. 

3.  Find  the  remaining  two  roots  by  §  207  or  208. 


$354,355.]      NUMBER  Off  ROOTS. — COEFFICIENTS.        271 

4.   One  root  of  the  equation, 

x3—  1  =  0,  i.  e.  x3  =  l, 
is,  obviously,  1.     What  are  the  other  roots  (§  207)  ? 

Arts.  i(-  1  +  (—  3)*),  and  \ (- 1  - (- 3)  *). 

d.)  Either  of  the  roots  of  the  last  equation,  being  cubed, 
will  produce  1.  Thus,  every  number  has  three  cube  roots ; 
one,  real;   and  two,  imaginary. 

In  like  manner,  every  number  has  four  fourth  roots  ; 
and,  in  general,  n  nth  roots. 

§  054.  e.)  The  principle  of  §  353  may  be  applied  to 
equations  of  fractional  degrees  (§  346.  2). 

Thus,  the  number  of  the  roots  of  the  equation, 

x'1  —  7icTj"  -j-  6  =  0,         may  be  said  to  be  _, . 

For  we  find  »*  =  1,  2,  or  -3  ; 

and,  consequently,  x  =  1,  4,  or  9. 

Now  these  three  values  of  x  correspond  to  six  values  of 

x'2,  only  three  of  which  satisfy  the  equation ;   as  will  be 

seen,  if  we  take  x  =  —  1,  —2,  or  +3.     The  values  of  x, 

therefore,  i.  e.  the  roots,  may  properly  be   said  to  be  half 

roots  (§12). 

i  i 

So,  the  equation,     x5  —  2  =  0,  i.  e.  Xs  =  2, 

obviously  gives         x~=.  8,  or  x  —  8  =  0. 

But  x-  8  =  (J-  2)(*M-  1  -V-3)(a£+- 1  -fV-0), 
only  one  of  which  partial  or  component  factors  (§  12),  with 
the  corresponding  partial  root,  is  found  in  the  given  equa- 
tion. The  equation  may,  therefore,  be  said  to  contain  only 
one  third  of  a  root.     See  §  221.  2,  3. 

COEFFICIENTS. 

§  355.   Let  av  a2  .  .  .  •  a*  be  the  roots  of  an  equation. 
Then  we  shall  have 


2  72 


THEORY  OF  EQUATIONS. 


[§  355. 


a?-\- A-^x"-1  .  .  -\-  An=  (x  —  ax)(x  —  a2) .  .  (x  —  a„)  =  0. 
Multiplying  (§283),  v*+\>Ax&~i  .  .  .  .  +  An  — 


xn  —  ax 

—  a2 

—  a. 


—  a, 


xn~- —  a,a0a, 

j.      ^      o 

—  axa2a^ 


a2a3a4 


&c. 


+  o.2an 
&c. 
Hence  (§  277), 

A±  =  — 0>\ — «2  —  a3    •    ■ 

^42  —  ala2-\-a1a3  .  .  -}-«!«„ 


xn~3  .  ,±a1a2  . .  a„ 


—  ct„ 


+  «2«rt  +  &(-'' 


J3  == — axa2a2  —  a1«2«4 


— axadaa 


Sec. 


^4  =  a1«2a3a4-(~aia2a3a5  "T~&c' 


Ar. 


±  o1a2aaa4«5  .  .  .  oa.  That  is. 

(1.)  The  coefficient  of  the  second  term  is  equal  to  the  sum 
of  the  roots  loith  their  signs  changed  (§  213.  3). 

(2.)  The  coefficient  of  the  third  term  is  equal  to  the  sum 
of  the  products  of  the  roots  taken  two  and  two  (§  213.  4); 
(3.)  that  of  the  fourth  term,  to  the  sum  of  their  products 
taken  three  and  three  ;  and  so  on,  the  signs  of  the  roots  be- 
ing changed  in  every  case. 

(4.)  The  absolute  term  (i.  e.  the  coefficient  of  x°  [§  208]) 
is  the  product  of  the  roots  taken  cdl  together,  with  their 
signs  changed. 

a.)  It  is  evident,  that,  in  the  third,  fifth,  seventh,  &e. 
terms,  the  number  of  factors  being, even,  the  result  will  be 
the  same,  whether  the  signs  of  the  roots  be  changed  or  not 
(§213.4). 

b.)  The  last  term  will  be  positive  or  negative,  according 
as  the  number  of  positive  roots  is  even  or  odd  (§215.  1,  2). 

c.)  If  the  roots  be  all  negative,  the  factors  will  be  of  the 


§  356.J     FORM  OF  THE  ROOTS. — NOT  FRACTIONAL.        273 

form  x-\-a1,  x-\-a2,  &c,  and  the  terms  will,  evidently,  all 
be  positive  (§  215.  1,  3) ;  if  the  roots  be  all  positive,  the 
terms  will  be  alternately  positive  and  negative  (§  215.  1,  3). 

d>)  If  the  coefficient  of  the  second  term  be  zero,  the  sum 
of  the  positive  roots  is  numerically  equal  to  the  sum  of  the 
negative  roots  (§  214.  1). 

e.)  Every  root  of  the  equation  is  a  divisor  of  the  last 
term;  and,  hence,  if  the  last  term  be  zero,  one  of  the  roots 
must  be  zero  (§  214.  2);  or  rather,  in  this  case,  the  equa- 
tion becomes  of  the  (n — l)th  degree  (§  203). 

1.  Form  the  equation,  whose  roots  are  2,  3,  and  — 4, 
Ans.  (x—2)(x  —  3)(x4-4:)  =  x^—xn—lix-\-2i=zO. 

2.  Form  the  equation,  whose  roots  are  1,  1,  2,  and  3. 

Ans.  x*  —  7x3  +  17x2  —  17  -f-  6  =  0. 

3.  Given  the  roots,  2,-1  +y—  3,-1  —  ./—  3  ;  to 
find  the  equation.  Ans.  x3  —  8  =  0. 

FORM  OF  THE  ROOTS. 

§356.  Let  the  equation,  st*-\- A^x"-1  .  .  -\-An  =  0,  have 
its  coefficients  all  integral  (the  coefficient  of  the  first  term 
being  unity) ;  it  is  required  to  determine  whether  it  can 
haye  a  fractional  root. 

If  possible,  let  -r,  a  fraction  in  its  lowest  term?,  be  a  root 

of  the  equation.     Then  we  shall  have 

an  an~ 1 

Multiplying  by  b"~i,  and  transposing, 

^=z-,A1an-^—A2an~^b  .  .  —  AJT~*. 

Now  all  the  terms  of  the  second  member  of  this  equation, 
are  whole  numbers,  while  the  first  member  is  an  irreduci- 
ble fraction.  That  is,  we  have  an  irreducible  fraction 
equal  to  a  whole  number;  which,  evidently,  is  impossible. 
Ilence, 


274  THEORY  OF  EQUATIONS.  [§35?. 

If  the  coefficient  of  the  first  term  he  unity,  and  the  other 
orfficients  all  integral,the  equation  cannot  have  a  fraction- 
al root. 

a.)  It  is  not,  therefore,  to  be  inferred,  that  all  the  root? 
are  integral.  They  may  be  either  integral,  irrational 
(§§  153,  175),  or  imaginary  (§  23./.  2). 

§  357.  Let  the  coefficients  of  the  equation,  X=  0,  be  all 
real ;  and  let  a  -\-  bj1—  1  be  a  root  of  the  equation. 

The  quantity  bj— 1  can  have  resulted  only  from  the 
extraction  of  an  even  root,  which  must  have  given,  at  the 
same  time,  —  bj— 1  (§  23./.  1).  Consequently,  a  —■  bj—l 
must  be  a  root  of  the  equation. 

Otherwise ;  the  sum  and  product  of  the  roots  (§  355.  1, 
4)  must  both  be  real.  Therefore,  if  one  root  be  a  +  fl«/— 1, 
another  must  be  a—  bj—  1,  so  that  their  product  (§  186) 
and  sum  may  both  be  free  from  imaginary  expressions. 
Hence, 

If  the  coefficients  of  an  equation  be  all  real,  the  number 
of  its  imaginary  roots  must  be  even  (§  217.  I.). 

0.)  Thus,  there  may  either  be  no  (§  63.  N.)  imaginary 
roots,  or  there  may  be  two,  four,  &c.     Hence, 

b.)  Cor.  r.  Every  equation  of  an  odd  degree  has  at  least 
one  real  root,  with  a  sign  (see  c.  below,)  different  from  that 
of  the  last  term  (i.  e.  of  the  coefficient  of  x°), 

c.)    We  have  [§§186,  162] 

(a  +  6y—  l)(a—  bj—  1)  z=a2-i-&0, 
.i  positive  quantity  (§  11.  N.).     Hence, 

Ccr.  ii.  If  all  the  roots  of  an  equation  are  imaginary, 
the  last  term  must  be  positive  (§  216).     Hence, 

Cor.  in.  Every  equation  of  an  even  degree,  whose  last 
term  is  negative,  has  at  least  tivo  real  roots  ;  one  positive, 
and  the  other  negative  (§§  68.  a;  215.  2). 

1.  Giver,  the  roots,  5,  3  -\-*/ —  4,  3  — «/ —  4  ;  to  form 
the  equation.  Ans.  a:3  —  1  lx2  +  13.x*  —  65  ■—.  0. 


§  358,  359.]  signs  of  the  roots.  275 

2.  Form  the  equation,  whose  roots  are   —  6  -{-  5«/ —  1, 
—  C  —  5y—  1,  1  -fV—  4,  and  1  — y—  4. 

Ans.  x*  +  10x3  —  42a:2  —  62a: +  305  =  0. 

3.  Form  the  equation,  whose  roots  are  2,  —2, 1-fV— 37 
and  1— y— 3.  .4ws.  a:4  —  2x3  -j-  8a:  — 16  =  0. 

§  358.  d.)  Again  (§  218.  h), 

(x-a-5y-l)(x-fl-fiy-l)  =  (a:  — a)2-H2; 
a  result  necessarily  positive  for  every  real  value  of  x.    Con- 
sequently, 

Cor.  iv.  (1.)  The  product  of  all  the  imaginary  factors  is 
positive  for  every  real  value  of  x.     Hence, 

(2.)  The  sign  of  the  j£rs£  member,  for  any  ?-caZ  value  of 
x,  depends  on  the  real  factors.     And, 

(3.)  If  all  the  roots  are  imaginary,  the  first  member  will 
be  positive  for  every  real  value  of  x. 

e.)  The  product, 
(a;  _  a  — y—  5) (a:  —  a  -+V— b)  —  x-  —  2ax  -f  a-  +  62, 
of  the  factors  corresponding  to  each  pair  of  imaginary 
roots,  or  conjugate?  roots,  as  they  are  sometimes  called,  is 
real.     Hence, 

Cor.  v.  Every  equation  may  be  resolved  into  real  fac- 
tors ;  of  the  first  degree,  corresponding  to  the  real  roots  ; 
and  of  the  second  degree,  corresponding  to  each  pair  of 
imaginary  roots. 

SIGNS  OF  THE  ROOTS. 

§  359.   Let  a  be  a  root  of  the  equation, 
x'l  +  A1xn-lJrA2xn~2  .  .  .  .  -f  An^1x-\-An  =  6.  [{I) 

Changing  the  signs  of  the  alternate  terms,  we  have 
xn—  A  jK"-  1Jr  A  2xn-z—  A^xn-3-\-  &c.  —  0  ;        (2) 
or  (§  44.  a),  changing  all  the  signs  of  (2), 

—  xn+A1xn-1—A2xn---\-A3x'*-3—&c.  =  0.       (3) 

(e)  Lat.  conjugo,  to  join  together. 


27G  THEORY  0?  EQUATIONS.  [§  360. 

The  equations  (2)  and  (3)  are,  obviously,  the  same;  as 
will  be  seen  by  transposing,  in  each,  all  the  negative  terms 
to  the  other  side. 

Now,  if  -[-a  be  substituted  for  x  in  (1),  and  — a,  in  (2) 
when  n  is  an  even  number,  or  in  (3)  when  n  is  odd,  the  re- 
sults will  be  precisely  alike.  But  the  substitution  of  -f-  a 
in  (1)  reduces  the  first  member  to  0.  Consequently,  the 
substitution  of  —  a  in  (2)  or  (3)  reduces  the  first  member 
to  0,  and  therefore  —  a  is  a  root  of  the  equations  (2)  and 
(3).     Hence, 

If  the  signs  of  the  alternate  terms  in  an  equation  be 
changed,  the  signs  of  all  the  roots  will  be  changed. 

Form  the  equations,  whose  roots  are  1,  2,  and  3 ;  and 
— 1,  — 2,  and  — 3. 

§  3G0.  A  permanence*  o£  signs  occurs  when  two  succes- 
sive terms  are  affected  each  by  the  same  sign ;  a  variation, 
when  their  signs  are  different.  Thus,  x  -f-  a  =  0  -exhibits 
a  permanence,  and  x  —  a  =  0,  a  variation  ;  the  first  corres- 
ponding to  a  negative,  and  the  second,  to  a  positive  root. 

I.  Let  the  signs  of  the  terms  in  their  order,  in  any  com- 
plete equation  be  -\-  -| ( ,  and  let  a  new  factor, 

x  —  a  =.  0,  corresponding  to  a  new  positive  root,  be  intro- 
duced.    The  signs  will  be  as  follows,  viz, 

+  +  —  -  +  - 

+  ~ 

+  + +  ~ 


Now,  in  this  result,  it  is  manifest,  that  each  permanence 
is  changed  into  an  ambiguity ;  and  that,  whether  there  be 
one,  or  any  greater  number,  of  double  signs,  the  single 
signs  immediately  preceding  and  following  are  always  un- 
like. Hence  the  number  of  permanences  may  be  dimin- 
ished, but  cannot  be  increased. 

« 

Hence,  the  number  of  signs  being  one  greater  than  be- 
(/)  Lat.  permaneo,  to  continue. 


§361.]  'SIGNS  OF  THE  ROOTS.  277 

fore,  the  number  of  variations  also  must  be  at  least  one 
greater. 

Now  the  equation,  x — a  =  0,  containing  one  positive  root, 
has  one  variation.  Consequently,  as  every  additional  pos- 
itive root  introduces,  at  least,  one  additional  variation, 

The  member  of  variations  can  never  be  less  than  the 
number  of  positive  roots. 

II.  (1.)  By  like  reasoning  it  can  be  shown,  that  the  in- 
troduction of  a  negative  root  (i.  e.  of  the  factor  x-\-a)  will 
introduce  at  least  one  permanence ;  and  that,  therefore, 

The  number  of  permanences  cannot  be  less  than  the 
number  of  negative  roots. 

(2.)  Or,  if  we  change  the  signs  of  the  alternate  terms,  the 
variations  will  evidently  become  permanences,  and  the  per- 
manences, variations ;  and  the  negative  roots  will,  at  the 
same  time,  become  positive  (§  359). 

But  the  number  of  variations  in  this  equation  cannot  be 
less  than  the  number  of  its  positive  roots.  Therefore,  the 
number  of  permanences  in  the  primitive  equation  cannot 
be  less  than  the  number  of  its  negative  roots. 

Hence,  universally,  in  a  complete  equation, 

§  361.  The  number  of  positive  roots  cannot  be  greater 
than  the  number  of  variations  of  sign  ;  nor  the  number 
of  negative  roots,  greater  than-the  number  of  perman- 
ences. 

Note.    A  complete  equation  of  the  rath  degree, 

*n  +  A*"-1 +An_1x  +  An=zQ, 

must,  obviously,  contain  n4-l  consecutive  powers  of  x;  and,  of 
course,  n  +  1  terms  (§§195,  196). 

1.  How  many  permanences  and  variations  in  the  equa- 
tion, whose  roots  are  2,  2,  and  —  5  ? 

Ans.   The  equation  is 

(x  —  2)(x  —  2)(x-\-5)=:x3-\-x2  —  l6x  +  20  =  0; 
showing  one  permanence,  and  two  variations,  as  we  have 
seen  there  must  be. 

alg.         -  21 


-'fl  THEORY  OF  EQUATIONS.  [§362. 

2.    IIow  many  permanences  and  variations  in  the  equa- 
on,  whose  roots  are  1,  2,  4,  and  —  4  ? 

a.)  The  whole  number  of  variations  and  permanences 
must,  evidently,  be  equal  to  the  degree  of  the   equation 

e  equation  being  complete,  or,  if  not  complete,  being 
rendered  so  by  the  introduction  of  cyphers,  as  in  §  362). 

Therefore, 

Cor.  i.  If  the  roots  of  an  equation  be  all  real,  the  num- 
ber of  positive  roots  must  be  equal  to  the  number  of  varia- 
tions ;  and  the  number  of  negative  roots,  to  the  number  of 
permanences.     See  §  218.  1,  2,  3. 

§  362.  b.)  If  any  term  of  the  equation  be  wanting,  a  cy- 
pher may  be  put  in  its  place;  and,  obviously,  either  sign 
may  be  given  to  it  without  affecting  the  I'oots  of  the  equa- 
tion. 

Thus,  the  equation, 

x2  -f  25  =  0, 
may  be  written        x2  ±  0  -f-  25  =  0. 

Now,  in  this  equation,  if  the  upper  sign  be  taken  with 
;ie  middle  term,  there  will  be  no  variations ;  and,  of 
<  ourse,  the  equation  has  no  positive  root.  But,  if  the  low- 
er sign  be  taken,  there  will  be  no  permanences ;  and,  there- 
fore, the  equation  has  no  negative  root.  Consequently,  the 
roots  of  the  equation  are  imaginary  (§  353). 

So,  in  the  equation, 

x3  ±0  +  4^  +  7  =  0, 

be  upper  sign  be  taken  with  the  second  term,  there  will 
be  no  variation,  and  no  positive  root ;  and,  if  the  lower 
si<ni  be  taken,  there  will  be  but  one  permanence,  and,  of 
course,  not  more  than  one  negative  root.  The  other  two 
roots  are,  therefore,  imaginary. 

The  equations,  x-  ±  0  —  25  =  0, 

and  ^±0-4x4-7  =  0, 

exhibit  the  same  number  of  permanences  and  variations, 
whether  wc  take  the  upper  or  lower  sign  before  the  mis- 


§363,364]        SIGNS.  — IMAGINARY  ROOTS.  270 

sing  Jterm ;   and,  consequently,  it  cannot  be  inferred  that- 
the  roots  are  not  all  real. 
Hence, 

§363.  Cor.  ii.   If  the  introduction  of  +  0  in  place  oi 
missing  term  gives  a  different  number  of  permanences  and 
variations  from  that  given  by  the  introduction  of  —  0,  the 
equation  contains  imaginary  roots. 

c.)  This  will,  obviously,  happen,  if  the  terms  immediate- 
ly preceding  and  following  the  deficient  term  have  like  sign-. 

d.)  Also,  if  tico  or  more  successive  terms  be  wanting, 
then,  supplying  the  terms,  the  first  of  the  supplied  term-; 
may  always  have  the  same  3ign  as  the  term  following  all 
the  deficient  terms.  Consequently,  the  equation  must  have 
imaginary  roots. 

Thus,  in  the  equation, 

x3  —  0±0 — 1  =  0, 

if  we  take  the  upper  sign  before  the  third  term,  we  have 
three  variations,  to  which  negative  roots  cannot  correspond : 
if  we  take  the  lower  sign,  we  have  two  permanences,  to 
which  positive  roots  cannot  correspond.  Two  of  the  root.-, 
then,  can  be  neither  positive  nor  negative ;  and  must,  of 
course,  be  imaginary. 

§  364.  e.)  It  is  evident  also,  that,  the  greater  the  num- 
ber of  deficient  terms,  the  greater  difference  can  be  made 
between  the  numbers  of  variations  and  of  permanences, 
respectively;  and,  therefore,  the  greater  will  be  the  num- 
ber of  imaginary  roots  of  which  we  shall  be  assured.  Thus, 
it  is  easily  seen,  that, 

(I.)  If  an  odd  number  (2m-f-l)  of  consecutive  terms  be 
wanting,  the  number  of  imaginary  roots  must  be  at  least 
2m  -{-  2,  if  the  signs  of  the  terms  immediately  preceding 
and  following  the  deficient  terms  be  like;  and  at  least  2m. 
if  they  be  unlike. 

a.)  Thus,  in  the  equation, 

xA  ±  0  ±  0  ±  0  + 1  =  0, 


280  THEOKY  OF  EQUATIONS.  [§  365. 

we  find,  if  we  take  the  upper  6igns  throughout,  no  varia- 
tions ;  and,  if  we  take,  alternately,  the  lower  and  upper 
signs,  no  permanences.  Hence,  there  must  be  4(=  2m-\-2) 
imaginary  roots. 

b.)  In  the  equation, 

x±  —  0  —  0±0  — 1=0, 
we  may  have  one  permanence  and  three  variations,  or  one 
variation  and   three  permanences.     Hence,  we  may  have 
one  positive  and  one  negative  root ;  and  must  have  2(=  2m) 
imaginary  roots. 

(2.)  Also,  the  deficiency  of  an  even  number  (2m)  of  con- 
secutive terms  indicates  at  least  2m  imaginary  roots. 

c.)  Examine  the  permanences  and  variations  of  the  equa- 
tion,1 

x*  —  0±0  —  1  =  0. 

Notes.  (1.)  Giving  to  the  first  cypher  in  the  last  example,  and 
to  the  first  two  in  the  last  but  one,  the  sign  of  the  term  following  them 
nil,  we  have  an  odd  number  (2»i  —  1)  of  terms  wanting,  preceded 
and  followed  by  terms  of  like  signs.  Wherefore,  by  1,  above,  there 
must  be  at  least  2m{—2m  — 1+1)  imaginary  roots. 

(2.)  It  should  be  remembered,  that  there  may  be  more  imaginary 
roots  than  are  thus  indicated ;  and  that  there  are  frequently  imagi- 
nary roots  when  no  terms  are  wanting  (§§  216;  218.  4). 

TRANSFORMATION. 

§  365.   Let  it  be  required  to  transform  the  equation 
X=xn^-A1x"-i-{-A2xn-2  .  .  +  AH-1x  +  An=0,  (1) 

into  another  whose  roots  shall  be  less  than  those  of  the 
given  equation  by  x' . 

The  roots  of  the  new  equation  will,  of  course,  be  equal 
to  x  —  xf.  Let  y  =  x  —  x'.  Then  y  -\~  x!  ==  x  ;  and,  if  we 
substitute  y  -\-  x'  for  x,  we  shall  have  a  polynomial  of  the 
same  value  as  before,  but  expressed  in  terms  of  y(=x — x'} 
instead  of  a*.     Thus, 

X=(y+tf)*+A1(y+tfy-i-t-A^+s')-* , 

+  An_2(y+x<y  +  A^^y  +  x')  +  An=  0, 


§  366.] 

Developing, 


TRANSFORMATION. 


281 


yn  -\-  nx? 


yn-i -\-rix"1-1 

Jr{n  —  l)Alx/n~2 


rJn 


-\-A^n~- 


►=f     2 


[-2An_2x>   +An^xJ' 

+  An) 

Or,  putting  B1,  B.2,  &c.,  for  the  coefficients  of  y  i_1  , 
yn~2,  &c. 
X^yn-t-BsjO-i  +  B^-s  ...+Bn_ly  +  Bn  =  U.(:; 

Or,  again, 
X  —  (x—x/)n-\-Bl  (x—x'y-1  ..  +Bn- ,  (x— xJ)-\-Bn  =  0 ;  (4 
where  x — x1  may  be  regarded  as  the  unknown  quantity. 

Now,  obviously,  the  roots  of  (2),  (3)  and  (4)  are  the  val- 
ues of  y(—  x  —  x/);  and  are,  therefore,  less  by  x'  than  the 
roots  of  the  given  equation  (i.  e.  the  values  of  x). 

Hence,  the  transformation  required  is  effected  by  the 
substitution  of  y-\-x'  (i.  e.  of  \_x  —  x'~\-\-x')  for  x  in  the 
given  equation.     Thus, 

Find  an  equation,  whose  roots  shall  be  less  by  2  than 
those  of  the  equation, 

x2  —  9cc  +  20z=0. 
Substitute  y-\-2  for  x. 

Then  (y  +  2)  =  -%  +  2)  +  20  =  0, 


or         #2  +  4 
—  9 


■5y+6  =  0, 


y+    2- 

—  9X2^=0,     ovy' 
+  20 

is  the  equation  required,  whose  roots  will   be  found  to  be 
less  by  2  than  those  of  the  given  equation. 

§  366.  a.)    The  labor  of  effecting  this  substitution  may 
be  greatly  abridged,  especially  in  the  higher  equations. 

*24 


282  THEORY  OF  EQUATIONS.  [§  366. 

For  Bn,  i.  e.  the  coefficient  of  y°  in  the  transformed 
equation  (2),  is  simply  what  the  first  member  of  the  given 
equation  becomes,  when  x/  is  substituted  for  x.     That  is, 

Bn=f(x'). 

Bn_1  is  formed  by  multiplying  each  term  of  Bn  by  the  ex- 
ponent of  x1  in  that  term,  and  diminishing  the  exponent  by 
unity. 

Bn^2  is  formed  by  multiplying  each  term  of  Bn_x  by  its 
exponent  of  xf,  diminishing  the  exponent  by  unity,  and  di- 
viding by  2  ;  and  so  on. 

b.)  In  other  words,  each  term  of  Bn^x  is  the  first  de- 
rived function  (§  292.  N.  3)  of  the  corresponding  term  of 
Bn;  i.  e.  of  /(a/). 

Each  term  of  Bn_2  is  half  the  first  derivative  of  the  cor- 
responding term  of  BH^1 ;  i.  e.  half  the  second  (§  292.  N.  4) 
derivative  of  the  corresponding  term  of  Bn. 

So,  each  term  of  -S„_3  is  one  third  of  the  first  derivative 
of  the  corresponding  term  of  Bn-2  ;  i.  e.  one  sixth  of  the 
third  (§  292.  N.  4)  derivative  of  the  term  of  Bn. 

c.)  Hence,  Bn^1  is  called  the  first  derived  polynomial  of 
Bn,  or  of  the  given  equation ;   and  may  be  expressed  by 

-S},orby/'«>. 

I?n_3  is  half  the  second  derived  polynomial  of  the  equa- 

B"  f'ip^) 

tion,  and  may  be  expressed  by  —-,  or  by' — - — . 

bo,  2*„_3  — 2-3  —  ~273    »     ^«-4  —  2.3.4      2.3.4' 

^-^-2.3.4.5-2.3.4.5'  ^C' 

1.   Diminish  by  2  the  roots  of  the  equation, 

xi  _}_  5X  -j-  6  =  0. 
The  transformed  equation  will  be  of  the  form.,  ■ 


§367.]  TRANSFORMATION. — COEFFICIENTS.  233 

And  we  shall  have 
B2=f(xf)  =zxf2  +  5x,+  6  =  22-\-5x2  +  G  =  20; 
B1  =  B2'=f'(xf)  =  2x'  +  5  =  2X2  +  5  —  9; 
B0  =  *B2"  =  1X2  =  1  {B0  denoting  the  coefficient  of  ,y2). 
#2  ~\~  9y-f"20  =  0  is  the  equation  required. 

§  367.   d.)   A  still  more  convenient  method  of  finding 
these  coefficients  results  from  the  form  of  equation  (4). 
For,  comparing  (4)  and  (1),  we  have 

(x—x'y-\-Bf(x  —  x'y-i- +Bn_^(x  —  x')  +  BK 

-xn-\-Axxn-^ -\-An-xx-\-At.  (5) 

Now  every  term  of  the  first  memher  of  this  equation  is 
divisible  by  x  —  x',  except  the  last  term,  Bn ;  which  will  be 
the  remainder. 

In  like  manner,  every  term  of  the  quotient  which  results 
from  this  division  is,  evidently,  divisible  by  x  —  x',  except 
the  last,  Bn_x,  which  will  be  the  remainder;  and  so  on. 

But  the  second  member  being  absolutely  (§  37.  d)  equal 
to  the  first,  its  successive  divisions  by  a:  —  x1  must  result  in 
the  same  quotients  and  remainders  as  the  division  of  the 
first  member. 

Hence, 

If  we  divide  the  given  equation  by  x — xJ,  the  remainder 
will  be  J?,',  the  coefficient  of  y°  in  the  transformed  equation. 

If  we  divide  the  resulting  quotient  by  x  —  a/,  the  re- 
mainder will  be  B'n_v  the  coefficient  of  y1 ;  and  so  on,  each 
of  the  coefficients  being  formed  by  the  successive  division 
of  the  several  quotients  by  a;  —  xf. 

e.)  It  is  evident  also  from  §  348.  b,  that  the  first  remain- 
der will  be  Bn  [=f(x')~]  ;  i.  e.  what  X  becomes,  when  x' 
is  substituted  for  x  (§  350.  2,  3,  4). 

1.  Transform  the  equation,  x-  -j-  9a:  +  20  =  0,  into 
another  whose  roots  shall  be  less  by  5  than  those  of  thf> 
given  equation. 


284  THEORY  OP  EQUATIONS.  [§  367 


:e'  +  9.r  +  20 

x  —  5 

Or  (§86),     l+9-|-20 

1 

x*  —  bx 

x  +  14 

x  —  5                   +  5  +  70 

5 

14x              x-\-    5 

1                       l  +  14,  +  90,2?„. 

\Ax  —  70 

19: 

=  -B„-r                +    5 

90  =  £„.  l,  +  19,£n_r 

y2  ~h  1%  +  90  =  0  is  the  equation  required. 
2.    Find  an  equation,  whose  roots  shall  be  less  by  3  than 
those  of  the  equation, 

k3  +  10x2  —  15a  +  30  =  0. 

Neither  the  first  coefficient  of  the  divisor  (§  350.  2), 
which  is  always  1,  nor  the  first  coefficients  of  the  quotients, 
each  of  which  is  the  same  as  the  first  coefficient  of  the  div- 
idend, need  be  written.     Thus, 

1  +  10  —  15  +    30  (3 

+    3  +39  +   72 
-+13-  +  24,  +  102  =  Bn=B3. 

+   3+48 

+  16,  +  72  =  Bft_1  =  Ba. 
3 


+  19  =  Ba_2  =  B1. 

.-.    yz  + 19?/2  +  12y  + 102  =  0  is  the  equation  required. 

3.  Find  an  equation,  whose  roots  shall  be  less   by  —  2 
(i.  e.  greater  by  2),  than  those  of  the  equation, 

x3  +  8a:2  —  20*  +  25  =  0. 
"We  must  here,  of  course  divide  by  x  —  ( —  2)  ;  i.  e.  by 
x  +  2.  Ans.  y3  +  2y2  —  M)y  +  89  =  0. 

4.  Find  the  equation  whose  roots  are   less   by  1  than 
those  of  the  equation, 

x*  —  2x--\-3x  —  4  =  0. 

Ans.  y3  +y-  +  2y  —  2  =  0. 

5.  Find  the  equation,  whose  roots  are  less   by  2  than 
those  of  the  equation, 

X5  +  2x3  _  6x2  — 10*  +  3  =  0. 
Ans.  y5  +  l(ty*  +  42y3  +  86y2  +  10y  +12  =  0. 


§  367.J      TRANSFORMATION. — CHANGE  OF  ROOTS.  285 

6.    Diminish  by  2.8  the  roots  of  the  equation, 
x*  —  12a;2  +  12a;  —  3  =  0. 

We  may  here  either  diminish  the  roots  of  the  equation 
by  2,  and  then  the  roots  of  that  equation  by  .8,  or  we  may 
diminish  the  roots  of  the  given  equation  at  once  by  2.8. 
The   former   method   is   generally  the  more  convenient, 

Thus, 

1+0—  12  +  12  —    3  (2 

2+    4  — 16  —    8 


2—    8—    4,- 

2+    8  *     0 

-11 

4         0,-4 
2  +12 

6, +  12 
2 

8+12—4 

-11         (.8 

.8  +    7.04  +  15.232 

+  8.9856 

8.8  +19.04  +11.232, 

—  2.0144 

.8  +   7.G8  +21.370 

9.6  +  26.72,  +  32.608 
.8  +    8.32 

10.4,  +  35.04 

■ 

8 

11.2 

Diminishing  the  roots  by  2,  we  find  the  equation, 

y4+83/3  +  12y2-5y-ll  =  0. 
Diminishing  the  roots  of  this  equation  by  .8,  we  have 
y*  + \l.2y3  +  35.04y2  +  32.608y  —  2.0144  —  0  5 
the  equation  required. 

7.  Diminish  by  1.3  the  roots  of  the  equation, 

x3  —  7a; +  7  =  0. 
An*.  x3  +  3.9a;2  —  1.93*  +  .097  =  0. 

8.  Diminish  by  14  the  roots  of  the  equation, 

x3  —  17a;2  +  54a;  —  350  =  0. 

Ans.  x3  +  25a;3  +  166a;  — 182  —  0, 


286  THEORY  OF  EQUATIONS.  [§  368. 

§  368.  If  the  coefficient  of  any  power  of  y  in  equation  (2) 
of  §365  reduce  to  zero,  that  term  will  be  wanting  in  the 
new  equation.     Thus  the  second  term  will  disappear  from 

the  equation,  if  nx!  +  A  z=  0 ;  i.  e.  if  x'  — .     Hence. 

n 

To  make  the  second  term  disappear,  we  must  make  x/  =: 

;  i.  e.  we  must  diminish  the  roots  by ;  or,  which 

n  n 

is  the  same  thing,  increase  them  by  -\ . 

lb 

a.)  This  will  be  evident  otherwise ;  thus, 

The  sum  of  the  n  roots  of  the  primitive  equation  is  —  A 

A 

(<$  355.  1).     Now  if  each  of  the  roots  be  increased  by  — , 

v  y  n 

their  sum  will  be  increased  by  A ;  and  will,  of  course,  be 
equal  to  —  A  +  A  =  0. 

1.   Remove  the  second  term  from  the  equation, 
Xi  _  4X3  _  19X2  _|_  106a;  —  120. 

Here  we  have  n  =  4,  and  i  =  —  4. 

.*.  x,  — —       —  — x; 

n  4 

and  we  must  diminish  the  roots  of  the  equation  by  1. 
1—4—  19  +  106  —  120  (1 
1—    3  —  22+84 
_  3  —  22  +  SlT-17!^  =  54. 

1  —     2  —  24 
—  2—  247+  60  =  #3. 
1—1 


—  1,-25  =  J52. 
1_ 

y*  —  25y  -  +  GQy  —  36  =  0  is  the  equation  required. 

Transform  the  following  equations  in  like  manner. 
2.   x*  —  3x2  —  4X  +  12  =  0. 

Ans.  x3  —  7x  +  6  =  0. 


§  369.]      TRANSFORMATION. — REMOVAL  OF  TERMS.        287 

3.  a^-f.  14x4  + 12a;3  —  20x2  +  14x  —  25  =  0. 

Am.  y*  —  78*/3  +  412?/2  —  757y-f-401  =  0. 

4.  x2-{-2^  +  52  =  0.     Am.  y*-\-(q*—p*)  =  0. 

6.)   The  last  result  leads  to  the  common  solution  of  the 
equation.     For,  by  transposition, 

if-  =  pn~  —  q2  ;   and.%y=±Q>2  —  q2)k 


But  ^  — a?-|-^. 


i 


x+p  =  ±(j>*.-q*)\ 

x  =  — ^>  ±  (jp2  —  y*)  ■ 

c.)  If  we  would  remove  any  other  term  from  the  equa- 
tion, we  must  make  the  coefficient  of  that  term  in  (2)  of 
§  365  equal  to  zero,  and  find  the  corresponding  values  of 
x1.  By  the  substitution  of  a  value  so  found  for  x1,  that 
term  will,  of  course,  vanish. 

It  is  obvious,  that,  to  remove  the  third  term,  we  must 
solve  an  equation  of  the  second  degree  ;  for  the  fourth,  one 
of  the  third  degree,  and  so  on. 

To  remove  the  last  term,  we  must  solve  an  equation  of 
the  wth  degree ;  in  fact,  the  given  equation  itself,  with  x1 
substituted  for  x.  The  values  of  x'  found  from  this  equa- 
tion will,  therefore,  evidently  be  the  roots  of  the  given 
equation. 

§  369.   If,  in  the  general  equation, 

xn-\-A1xn-l  +  A2xn-2  ....  +An_lx-{-An=0, 

we  put  y  =  rax  (i.  e.  substitute  —  for  x),  we  shall  have 

m 

£+^S£ +A-^+A"=°-- 

or  (§  46)  yn-\-Almyn-x  .  .  -\-An-lmn-xy  -\-Anmn  —  0 ; 
an  equation  whose  roots  are  in  times  those  of  the  primitive 
equation.     Hence, 

An  equation  will  be  transformed  into  another,  whose 
roots  shall  be  equal  to  the  roots  of  the  first  multiplied  by  any 


288  THEORY  OF  EQUATIONS.  [§  370. 

number,  as  m,  if  we  multiply  the  second  term  of  the  given 
equation  by  m,  the  third  by  ?n2,  and  so  on.     Hence, 

Cor.  i.  An  equation  having  fractional  coefficients  may 
be  changed  into  another  with  integral  coefficients,  by  trans- 
forming it  so  that  its  roots  shall  be  those  of  the  given  equa- 
tion multiplied  by  a  common  multiple  of  the  denominators. 

Cor.  II.  If  the  coefficients  of  the  second,  third,  &c.  terms 
of  an  equation  be  respectively  divisible  by  m,  m2,  &c,  then 
the  roots  of  the  equation  are  of- the  form  mx,  and  conse- 
quently m  is  a  common  measure  of  them. 

1.  Transform  the  equation, 

3x3-f4x2  — 5x+6  — 0, 

into  another  whose  roots  shall  be  three  times  those  of  the 
given  equation. 

Here  m  =  3.     .*.  y  =  Sx,  and  x  =  iy. 

Am.  3y3  +  I2y2  —  45y  +  162  =  0  ; 
or,  y3  +  4y3  — 15y  +  54  =  0. 

2.  Transform  the  equation, 

into  an  equation  with  integral  coefficients. 

Am.  x^  -f  8x2  + 108*  —  4320  =  0. 

§  370.  If  in  the  general  equation, 

*l  +  Ala+-i+Aax*-*  ....  +An_1x  +  An=0, 

we  substitute  -  for  x,  we  shall  have 

y 

^  +  A~  +  ^^ +An_11-  +  An=0; 

or,  clearing  of  fractions,  and  reversing  the  order  of  the 
terms, 

A^+A^^-i +  ^2y»+.41y  +  l=0; 

an  equation,  whose  roots  are  the  reciprocals  of  the  roots  of 
the  given  equation.     Hence, 

To  transform  an  equation  into  another,  whose  roots  shall 


§372.]  RECURRING  EQUATIONS.  289 

be  the  reciprocals  of  the  roots  of  the  first,  we  have  only  to 
reverse  the  order  of  the  coefficients. 

a.)  Cor.  We  may  also,  evidently,  transform  an  equation 
into  another,  whose  roots  shall  be  greater  or  less  than  the 
reciprocals  of  the  roots  of  the  given  equation,  or  multiples 
of  those  reciprocals,  by  applying  the  processes  of  §$  367, 
369  to  the  coefficients  taken  in  a  reverse  order. 

b.)  It  may  happen,  that  the  coefficients,  when  taken  in 
the  reverse  order,  shall  be  the  same  as  when  taken  direct- 
ly. In  such  a  case,  the  transformed  will  obviously  be 
identical  with  the  given  equation ;  and  will  have  the  same 
roots.  Consequently,  as  the  roots  of  the  transformed  are 
the  reciprocals  of  those  of  the  given  equation,  and,  at  the 
same  time,  are  identical  with  them,  one  half  of  the  roots  of 
the  given  equation  must  be  the  reciprocals  of  the  other  half. 

Thus  the  roots  will  be  a.  - ;   b,    T  '•>  &c- 

a  b 

c.)  If  the  coefficients  of  corresponding  terms  are  numer- 
ically equal,  but  have  unlike  signs,  the  same  is  true  of  the 
roots,  in  every  equation  of  an  odd  degree ;  and,  in  every 
one  of  an  even  degree,  whose  middle  term  is  wanting.  For, 
in  both  these  cases,  if  all  the  signs  of  the  transformed  equa- 
tion be  changed,  (which  will  not  affect  the  iralue  of  the 
roots,)  the  transformed  will  be  identical  with  the  primitive 
equation. 

§  371.   d.)    Such  equations  (§  370.  b,  c),  which  remain 

the  same,  when  -  is  substituted  for  x,  are  called  recurring9 
x 

or  reciprocal  equations. 

e.)  The  general  form  of  a  recurring  equation  is,  obvious- 

iy,  ' 

xn-\-A1xn~l+A2xn-2 +  A  ^-^A^x  -+-1  =  0. 

Eecurring  equations  have  certain  peculiar  properties, 
which  will  be  considered  hereafter. 

(g)  Lat.  recurro,  to  run  back. 
ALG.  25 


290  THEORY  OF  EQUATIONS.  [§  372,  873. 

LIMITS  OF  THE  ROOTS. 

72.   In  the  equation, 

(x —  «x)(^  —  Q>$)(x  —  «3)  ...   =  0, 
let  a1,  a2,  a3,  &c.  be  the  real  roots,  taken  in  the  order  of 
{heir  magnitude;  i..e.  a1>a2,  a2>a3,  &c. 

If  now  blt  >  av  be  substituted  for  x,  we  have 
(b1~al)(b1—a.2)(bl—a3)  .  .  .   (bl—an),    positive; 
all  the  real  factors  being  positive  (§§  68.  a;  358.  1,  2). 

If  b.2,  <  ax  and  >  a2,  be  substituted  for  x,  we  have 
(h  —  «i)(52  —  «2)(*2  —  «3)  •  •  •  (&2  —  «»)»    negative- 
one  of  the  real  factors  being  negative  (§  68.  a). 

So,  if  we  substitute  b3,  <a2  and  >  a3,  the  product  will 
be  positive  ;  two  of  the  real  factors  being  negative,  and  the 
rest,  positive. 

In  like  manner,  the  substitution  of  54,  <a3  and  >«4, 
will  give  a  negative  ;  of  &5,  <a4  and  >«5,  a  positive  re- 
sult ;  and  so  on.     Hence, 

(1.)  If  a  quantity,  greater  tlian  the  greatest  real  root  of  an 
equation,  be  substituted  for  x,  the  result  will  be  positive  : 
and, 

(2.)  If  quantities  intermediate  between  the  roots,  begin- 
ning with  the  greatest,  be  successively  substituted  for  x, 
the  results  will  be  alternately  negative  and  positive. 

The  roots  of  the  equation, 

X3  _  5X2  _|_  2x  +  8  =  0, 
are  4,  2,  and  —1.     Substitute  5,  3,  1,  0,  and  —2  fur  x, 
and  observe  the  signs  of  the  results. 

§  373.  a.)  Hence, 

Cor.  i.  When  two  quantities  are  successively  substituted 
for  x,  if  the  results  have  like  signs,  there  is  an  even  ;  if  un- 
like signs,  an  odd  number  of  real  roots  between  those  quan- 
tities. 

Note.    The  even  number  may  be  0  (§63.  N.). 


<^37-l,  375.]  LIMITS  OF  THE  ROOTS.  201 

b.)  From  1,  and  Cor.  i.,  it  is  evident,  that, 
Cor.  ii.  IF  a  number  less  than  the  least  real  root  be  sub- 
stituted for  x,  the  result  will  be  positive  or  negative,  aecord- 
iug  as  the  number  of  real  roots  is  even  or  odd. 

c.)  If  the  degree  of  the  equation  be  odd,  the  substitution 
of -j-  oo  for  x  will  render  the  first  member  ■positive  ;  and  of 
—  oo,  negative.     Hence  (§  373.  a), 

Cor.  in.  (1.)  Every  equation  of  an  odd  degree  must  have 
at  least  one  real  root  (§  357.  b)  ;  and  (2.)  the  whole  number 
of  its  real  roots  must  be  odd. 

d.)  If  the  degree  be  even,  and  the  last  term  negative,  the 
substitution  either  of  -\-  oo  or  of  —  oo  will  render  the  first 
member  positive  ;  and  the  substitution  of  0  will  render  it 
negative.     Hence, 

Cor.  iv.  (1.)  Every  equation  of  an  even  degree  has  an 
even  (§373.  JST.)  number  of  real  roots;  and  (2.)  every  equa- 
tion of  an  even  degree,  whose  last  term  is  negative,  has  at 
least  two  real  roots,  one  positive  and  the  other  negative 
;§  357.  Cor.  in.). 

§374.  e.)  If  the  substitution  of  p,  and  of  every  number 
greater  than  p,  renders  the  result  positive,  then  p  is  great- 
er than  the  greatest  real  root ;  and  is  called  a  superior  li?n- 
it  of  ^the  roots. 

f.)  So,  if,  the  signs  of  the  alternate  terms  being  changed 
(§359),  the  substitution  of  q,  and  of  every  number  greater 
than  q,  renders  the  result  positive,  then  — q  is  less  than  the 
least  real  root  (i.  e.  it  is  an  inferior  limit). 

§375.  Let  Ah  be  the  first,  and  Am,  numerically  the 
greatest,  negative  coefficient  of  any  complete  (§3G1.  a) 
equation, 

xn  +  i/-i  .  .  —  Ahxn~h  .  .  —Amx"-m  .  .  -f  An=  0 

Now,  if  all  the  coefficients  after  Ah  be  negative,  the  sum 
of  those  terms  will  be  numerically  equal  to  the  sum  of  the 
preceding,  positive  terms. 


592  THEORY  OF  EQUATIONS.  [§  376. 

Consequently,  any  value  of  x,  which  renders  the  sum  of 
the  preceding  positive  terms  numerically  greater  than  the 
sum  of  the  negative  terms,  is  a  superior  limit  of  the  roots. 

And,  with  still  greater  reason,  any  value  of  x,  which 
renders  xn  numerically  greater  than  the  sum  of  the  nega- 
tive terms,  is  a  superior  limit. 

The  most  unfavorable  case  possible  is,  evidently,  when 
all  the  coefficients  after  Ah  are  negative,  and  each  of  them 
is  equal  to  Am,  the  greatest. 

Any  value  of  x,  then,  which  makes 

x»>Jm(x'l-»  +  xn-h-i   .  .  .  +  *  +  l),  (1) 

or  (§261)  *>^«C^71).  <2) 

is  a  superior  limit. 

Now  (2)  will  certainly  be  ti-ue,  if  we  have 

x'i-M-i  x~<"-1} 

x"  >  Am ;    or  1  >  A, 


x_l  ■  ~"    x_i      > 

or  x-'<-i(.r_i)  >  Jm.  (3) 

But         x—  1  <x,  and  (x  —  l)*-1  <s*-i. 

Therefore  (3)  will  be  true,  if  we  have 

{x_1y-l(x_1)^(x_1)^=zAm.  (4J 

and,  with  still  greater  reason,  if  (x  —  l)h  >  Am.  (5) 

Also,  (4)  and  (5)  give    x—  1  =  ,  or  >  (^„,)*; 

or  x=z  ,or>(Jw)*  +  l.  (6) 

That  is,  in  a  complete  equation, 

§  376.  7f  we  increase  by  unity  that  root  of  the  greatest 
negative  coefficient,  ivhose  number  is  equal  to  the  number  of 
terms  preceding  the  Jirst  negative  term,  the  result  will  be  a 
superior  limit  of  the  roots. 

Find  superior  limits  of  the  roots  of  the  following  equa- 
tions. 

1.    x*  —  ox3  +  37x2  —  3x  -f  39  =  0. 


§  377.]  LIMITING  EQUATION.  293 

Here  Am  =  5,  and  h  —  1. 

1  i 

(Am)h  + 1  =  5T  +  1  =  6,  the  limit  required. 

a.)  If  the  second  coefficient  be  negative,  the  limit  found 
will  be  the  greatest  negative  coefficient  increased  by  unity. 

2.  aj«-f-7:K±--12a;3  — 49fc*-f-52se  — 13  =  0. 

Ans.  (49)^  +  1  =  8. 

3.  a^-fll*2  —  25*  —  67  =  0. 

.4/w.  (67)^+1  =  6. 

6.)  If  the  signs  of  the  alternate  terms  be  changed  (§  359), 
and  a  superior  limit  be  found,  that  limit  with  its  sign  chang- 
ed will  be  an  inferior  limit ;  or,  as  it  is  sometimes  called,  a 
superior  limit  of  the  negative  roots. 

c.)  A  number,  which  is  numerically  less  than  the  least 
positive  or  negative  root  is  sometimes  called  an  inferior 
limit  of  the  positive,  or  of  the  negative  roots. 

Let  the  equation  be  found,  whose  roots  are  the  recipro- 
cals of  the  roots  of  the  given  equation ;  and  let  the  superi- 
or limits  of  the  positive  and  negative  roots  of  this  new 
equation  be  found. 

Now  those  roots  of  the  new  equation,  which  are  numer- 
ically the  greatest,  are  the  reciprocals  of  those  of  the  given 
equation,  which  are  numerically  the  least. 

Therefore,  the  reciprocals  of  the  superior  limits  of  the 
positive  and  negative  roots  of  the  new  equation  will  be  the 
inferior  limits  of  the  positive  and  negative  roots  of  the 
given  equation. 

LIMITING,  OR  SEPARATING  EQUATION. 

§  377.  Let  al,  a„,  a3,  &c.  be  the  real  roots,  taken  in  the 
order  of  their  magnitude,  of  the  equation, 

3*+-41icB-i  .  .  .  +  An„1x  +  Jn=0.  (1) 

Diminishing  the  roots  of  this  equation  by  a/,  we  have 
(§365) 

*25 


m 


294  THEORY  OF  EQUATIONS.  [§  377. 

9*+Ma*r*  +  B#— +  BB_1y-^BH=0;      (2) 

in  which  ($§  365 ;  366.  c) 

J?n_1  -fix')  =  na"-i  +(«- 1)4^-2  .  .  J- J„_1.(3) 

Also,  -5,,-!,  is  the  sum  of  the  products  of  the  roots,  with 
their  signs  changed,  of  equation  (2)  (i.  e.  of  the  products  of 
x' —  av  x1 —  a2,  .  .  x1 —  a„),  taken  n  —  1  at  a  time  (§  355). 
That  is, 

B„_1  =  (x'—  a2){xJ—  a3){x>—  a4)  .  .  .  (x'—an)  + 
(x'—a1)(x'—a3)(x'—a4)  .  .  .  (x'—an)-\- 
(x'—al)(x!—a2)(x'—ai)  .  .  .  (x1—  o„)  +   j 

•  •  • 

(^ «i)(*'' a2)(X' a3)     •    •    •    (^ an-l)jJ 

each  term  consisting  of  n  —  1  factors  ;  and,  of  course,  each 
factor  being  found  in  every  term  but  one. 

If  now,  in  this  value  of  Bn^^  we  make  x* '  =  av  a2,  a3, 
&c,  successively,  we  shall  have  (§  68.  a) 
Bn.r  —  (al—a2)(a1—a3)(al—a4)  . .  —  +.+•+  .  .=  +  ; 
Bn.1  =  (a2— a1)(a2—a3)(a2— a4)  . .  =  —  .  +  ."+..=—  ; 

#«-,  =  («3— «x)(«3— «2)(«3— a4>  •  ■  —  —  •  —  ■  +  ••  =  +  ? 
&c. 

That  is,  if  we  substitute  a1,  a2,  a3,  &c.  for  x!  in  Ai-p 
the  results  are  alternately  positive  and  negative. 

Hence  (§  372),  the  real  roots  of  Bn_l  =.  0  lie  between 
ap  a2,  a3,  &c. ;  and  therefore,  putting  x  m  place  of  x1,  we 
have  the  equation,  Bll^1  ss 

na;"-!  +  (n— 1)  ^a*-2  +  (n— 2)A2xn~3  .  .  +A„^1  =  0  ; 
whose  real  roots  severally  lie  between  those  of  the  given  equa- 
tion ;  and  which  is  thence  called  the  separating  or  limiting 
equation. 

a.)  Bn-V  we  have  seen  (§  366),  is  the  first  derived  poly- 
nomial of  the  given  equation.     That  is, 
f(x)  =  0,or  X'  =  0 
is  the  limiting  equation  of 

f{x)  —  0,  or  A'  —  0. 


S3 
-\ 
§  378.]  equal  roots.  296 

Hence,  the  separating  or  limiting  equation  is  properly 
called  the  derived  equation. 

b.)  It  is  obvious,  that  iff(x)  —  0  have  real  roots  (as  as- 
sumed in  the  investigation),  the  greatest  and  least  are  res- 
pectively greater  and  less  than  the  greatest  and  least  real 
roots  of/' (a?)  =:  0. 

EQUAL  ROOTS. 

§  378.  c.)  If  the  given  equation  have  two  roots  equal,  an 
a<L=-av  the  factor  x  —  aY  will,  evidently,  be  found  in  each 
of  the  terms  of  Blt-l  [§  377.  (4)]  ;  and,  consequently,  when 
x  —  av  we  shall  have  B^^  [—/'(a?)]  —  0  ;  i.  e.  aL  will 
be  a  root  of/' (a;)  =  0. 

So,  if  a3  =  a2  =  av  the  factor  (x —  aj2  will  be  found 
in  each  of  the  terms  of  BH^X,  i.  e.  of  f'(x);  and  the 
equation,  f'(x)  =  0,  will  have  two  roots  equal  to  al ;  and 
so  on. 

d.)  On  the  other  hand  also,  it  is  evident,  that  no  factor 
can  exist  in  all  the  terms  of  2?„_i [=/'(#)],  unless  it  enter 
more  than  once  in  f(x),  i.  e.  in  the  given  equation ;  and, 
that,  if  a  factor  appear  any  number  of  times  in  /'(#).  it 
must  be  contained  once  oftener  in /(a;). 

e.)   Hence,  to  find  whether  an  equation  has  equal  roots, 

Form  the  derived  or  limiting  equation,  /'(a-)  =  0;  and 
find  the  greatest  common  divisor  (§  104),  D,  of  the  polyno- 
mials,/^) and  f'(x). 

Make  Z)  =  0,  and  find  its  l'oots.  Each  of  these  roots 
will  be  contained  once  oftener  in  the  primitive  equation, 
f(x)  —  0,  than  in  D  —  0. 

/.)  If ' f{x)  and/'(x)  have  no  common  divisor,  the  given 
equation,  f(x)  =z  0,  has,  of  course,  no  equal  roots. 

1.     Given    f(x)  =z  x3  —  4a;2  -f  5x  —  2  =  0,    to    find 
whether  it  has  equal  roots. 
The  derived  equation  is 

f'(x)=3x!i  —  8x  +  5  =0; 


296  THEORY  OF  EQUATIONS.  [§379. 

and  the  greatest  common  divisor  off(x)  and  f'(x)  is 

a: —  1. 
Hence,  there  are  two  roots  equal  to  1. 
Dividing/(x)  by  (*-l)2(§8G), 


1 

+  2 
—  1 


1  —  4-J-5  —  2 

•+2  —  4 

-1  +  2 


1-2, 

we  find,  for  the  remaining  factor,  x  —  2  =  0. 
Therefore,  the  roots  of f(x)  =  0  are  1,  1,  and  2. 

2.  Given  x±  —  6x3  +  13a;2  —  12a;  +  4  =  0,  to  find  the 
equal  roots,  if  there  be  any.  Ans.  1,  1,  2,  and  2. 

3.  Given  x3  +  5x2  +  3x  —  9  =  0,  to  find   the    equal 

roots.  Ans.  —  3,  and  —  3. 

Note.    Between  two  equal  roots  of  an  equation,  there  can  evi- 
dently be  no  intermediate  root,  unless  it  be  equal  to  each  of  them. 
Thus,  the  derivative  of  the  equation, 

x2  +  2px  +  q2  =  0,    is  2x  +  2p  =  0,  or  x  +p  =  0  ; 

and  the  root  of  this  derived  equation  is  — p,  which  lies  between  the 
two  roots  of  the  given  equation, 

— P+o/(P2—  q%     and  -p-j^-q«~). 

Now,  if  q  becomes  nearly  equal  to  p,  the  quantity  under  the  rad- 
ical becomes  small,  and  the  two  roots  become  nearly  equal.  Also, 
if  q  becomes  equal  to  jo,the  radical  disappears;  and  the  roots  become 
equal,  taking  the  form  x  =  — p  ±  0. 

Thus,  the  separating  root  is  always  intermediate  between  tho  un- 
equal roots;  and  is  the  limit  to  which  they  approach,  as  they  become 
equal. 

sturm's  theoresi. 

§  379.  Sturm's  Theorem  is  a  method,  discovered  by 
M.  Sturm  in  1829,  of  finding  the  exact  number,  and,  near- 
ly, the  situation,  of  the  real  roots  of  an  equation. 

The  number  of  the  real  roots  being  known,  the  number 
of  imaginary  roots  is  known  of  course  (§  353,  a). 


( 380,  381.]  sturm's  theorem.  297 

Let      X=x"  +  Alxn~1  ....  -\-An.xx-{-An—0 
be  an  equation  which  contains  no  equal  roots ;   and  let  X' 
represent  its  first  derived  polynomial  (§§  366.  c  ;  292.  N.  3). 

Let  also  the  process  of  finding  the  greatest  common  di- 
visor (§  104)  be  applied  to  the  polynomials,  X  and  X'.  as 
follows ;  viz. 

Divide  Xhy  X',  till  a  remainder  is  obtained  of  a  degree 
lower  than  X'. 

Change  all  the  signs  of  this  remainder,  and  represert 
the  resulting  quantity  by  X2. 

Divide  X'  by  X2,  change  the  signs  of  the  remainder, 
and  designate  the  result  by  X3. 

Continue  this  process,  always  changing  the  signs  of  (he 
remainders,  till  a  remainder  is  obtained  independent  of  x. 

Notes.  (1.)  This  last  remainder  will  not  be  zero  ;  because,  by 
hypothesis,  the  equation  does  not  contain  equal  roots;  and,  therefore, 
the  polynomials  .XT  and  X'  have  no  common  measure  (§§  104;  378. 

/)• 

(2.)  In  performing  these  divisions,  any  positive  factor  not  found 
in  one  of  the  polynomials  may  be  introduced  or  rejected,  in  the  other 
(§  100.  a). 

(8.)  The  numbers,  2,  3,  &c,  are  used  to  distinguish  the  func- 
tions, Xo,  X3,  &c.,  from  simple  derived  functions,  which  would  be 
more  appropriately  denoted  by  X",  X"',  &c  (§  292.  N.  4). 

§  380.  The  result  of  the  above  operations,  representing 
the  successive  quotients  by  Qx,  (?2,  &c,  may  be  expressed 
as  follows ;  viz. 

X=X'Q1-X2;         ] 
X'  =  X2Q2-X5; 

X2=X3Q3-X^,  (U 

.... 

1 

XH- 2  —  -A „_  j  v„_  j       A „  ;  J 

Xn  representing  the  final  remainder,  which  is  indepeji- 
dent  of  a:,  and,  as  we  have  seen,  not  equal  to  zero. 

§  381.  I.  Now,  obviously,  any  one  of  these  functiur.? 
may  become  equal  to  zero  for  particular  values  of  x.     We 


298  THEORY  OP  EQUATIONS.  §  382,  383.] 

must  inquire,  whether  two  consecutive  functions  can  become 
zero  at  the  same  time  ;  i.  e.  for  the  same  value  of  x. 

Suppose  that  X2  and  X3  become,  at  the  same  time,  equal 
to  zero.  Then,  making,  in  the  third  of  equations  (1), 
X2  =  0,  and  X3  —  0,  we  find  X4  =  0. 

So,  from  X3  ■=  0,  and  X4  =  0,  Ave  find  X5  =  0  ;  and  so 
on,  till,  from  the  last  equation,  we  find  Xn  ■=.  0,  which  is 
contrary  to  the  hypothesis. 

Or,  proceeding  in  like  manner  in  the  other  direction,  if 
X2  and  X3,  or  any  other  two  consecutive  functions  become 
zero  simultaneously,  there  must  also  result  at  the  same 
time,  X=z  0  and  X'  =.  0.  This,  again,  is  impossible,  be- 
cause the  roots  of  X'  ■=.  0  are  intermediate  between  those 
of  1=0  (§  377.  a) ;  and  moreover,  there  are  no  equal 
roots  (§  378.  c). 

Hence, 

No  two  consecutive  functions  of  the  series,  X,  X',X2,&c, 
can  become  zero  at  the  same  time ;  i.  e.  for  the  same  value 
of  x. 

§  382.   II.   Again,  let  any  one  of  the  functions,  as  X3 
become  zero  for  a  particular  value  of  x. 
Making  X3  =  0  in  the  equation, 

we  have  X2  =  —  X4.  Hence, 

If  a  particular  value  of  x  reduces  one  of  the  functions  to 
zero,  the  adjacent  functions  must  have  unlike  signs  for  that 
value  of  x. 

§  383.  Let  now  different  values,  as  p,  q,  &c,  be  substi- 
tuted for  x  in  the  functions,  X,  X',  X2,  X3,  &c. ;  and  let 
the  resulting  signs  of  the  several  functions  be  written  in 
order,  and  the  number  of  their  variations  be  noted. 

And,  in  the  first  place,  the  signs  of  the  functions  will  re- 
main unchanged,  and  the  number  of  their  variations,  of 
course,  unaffected,  so  long  as  q  is  less  than  the  least  (§  373. 
b)  real  root  of  the  equations} 


§  884.]  stdrm's  theorem.  299 

X  =  0,  X'  =  0,  X2  =  0,  X3  =  0,  &c. 

But,  if  q  becomes  equal  to  a  root  of  one  of  the  equations, 
the  corresponding  function  will  become  zero ;  and,  as  q  in- 
creases still  more,  the  function  will  appear  with  its  sign 
changed  (§  373.  a). 

We  must  inquire,  what  will  be  the  effect  of  this  change 
of  sign  on  the  order  of  the  signs,  and  on  the  number  of 
their  variations. 

§  384.  1.  First,  let  q  be  the  smallest  of  all  the  roots  of 
the  equations  (1),  and  let  it  be  a  root  of  one  of  the  auxiliary 
equations,  as  X3  =  0. 

Then  we  shall  have  X3  =  0,  and  X2  =  —  X4.  That  is, 
X2  and  X4  will  have  unlike  signs  (§  382),  Moreover, 
neither  of  them  can  become  zero  at  the  same  time  with  X3 
(§381). 

We  know  also,  that  neither  X2  nor  X4  can  have  chang- 
ed'its  sign  ;  because  we  have  not  passed  any  of  the  roots  of 
X„  r=  0,  or  X4  =  0,  q  being  the  least  of  all  the  roots. 

Therefore,  whatever  may  have  been  the  sign  of  X3,  be- 
fore it  became  zero,  the  signs  of  X2  and  X4  having  been 
unlike,  the  three  signs-  must  have  exhibited  one  variation 
and  one  permanence.     Thus,  they  must  have  been  either 
-\-  ±  — ,   or—  ±  +. 

If  now  we  substitute  for  x  a  quantity  greater  than  the 
least  root  of  X3  =  0,  and  less  than  the  least  root  of  X.2  =  0 
and  X4  — 0,  the  signs  of  X2  and  X4  will  remain  as  they 
were ;  while  the  sign  of  X3  will  be  changed  (§  373.  a). 

The  signs  will  then  stand  thus,  viz. 

+   :f  — ,    or—   q:  -f; 
still  showing  one  valuation  and  one  permanence,  as  before. 

The  same  reasoning,  obviously,  applies  to  any  function 
intermediate  between  X  and  X„.     Hence, 

The  substitution  of  a  root  of  an  intermediate  equation,  or 
a  change  of  sign  of  one  of  the  intermediate  functions  does 
not  affect  the  number  of  variations  of  sign  in  the  series. 


300  THEORY  OF  EQUATIONS.  [§  385. 

§  385.  2.  "We  need,  therefore,  to  consider  the  case  only, 
in  which  X  changes  its  sign  in  consequence  of  passing  a 
root  of  the  primitive  equation,  X=0. 

In  examining  this  case,  we  must  remember,  that,  if  X=0 
have  real  roots,  the  least  of  them  is  less  than  the  least  real 
root  of  X'  =■  0  ("S  377.  b) ;  also,  that  X  being  one  degree 
higher  than  X',  one  of  the  equations,  XrrrO  and  X'-=z(). 
must  have  an  odd,  and  the  other,  an  even  number  of  real 
roots  ($  373.  c,  d). 

Consequently,  when  we  substitute  for  x  a  quantity  less 
than  the  least  real  root  of  X=  0,  X  and  X'  must  have  un- 
like signs  (§  373.  b). 

But  X  changes  its  sign  in  passing  the  least  real  root  of 
X=  0.  If,  therefore,  we  substitute  for  x  a  quantity  greater 
than  that  least  root  of  Xm.  0,  and  less  than  the  least  root 
of  X'  =z  0,  X  and  X'  will  have  like  signs. 

That  is,  these  signs,  which  before  exhibited  a  variation. 
willjiow  exhibit  a  permanence.  « 

Therefore,  as  the  number  of  variations  in  the  other  func- 
tions has  undergone  no  change  (§  384), 

The  whole  number  of  variations  is  diminished  by  one, 
in  passing  a  real  root  of  X  =  0. 

a.)  The  same  reasoning  will  apply  to  the  next  real  root 
of  X=  0  ;  and  so  on. 

For  suppose,  that  we  have  passed  any  equal  number  of 
the  real  roots  of  X=  0  and  X'  =  0. 

Now,  if  we  substitute  for  x  a  quantity  less  than  the  next 
greater  root  of  X=0,  the  signs  of  X  and  X'  will  be  un- 
like (§  373.  b),  and  will  constitute  a  variation.  * 

But,  if  we  substitute  lor  x  a  quantity  greater  than  that 
next  root  of  X=0,  and  less  than  the  succeeding  root  of 
X'  —  0,  X  will  change  its  sign  ;  and  the  signs  of  X  and  X'. 
becoming  like,  will  constitute  a  permanence. 

b.)  In  fact,  after  we  pass  the  least  root  of  X=  0,  A'  and 
X'  have  like  signs,  till  we  pass  the  least  root  of  X'  =  0  ; 
when  they  become  unlike,  without  however  producing  an 


§  386,  387.]  sturm's  tiieorkm.  301 

additional  variation  (§  384).     Then,  in  passing  the  next 
root  of  X=0,  the  change  of  sign  in  X  introduces  a  per- 
manence instead  of  a  variation  (§  385). 
Hence, 

§  386.  If  two  quantities,  p  and  q  be  successively  substi- 
tuted for  x  in  the  functions,  X,  X',  Xa,  &c., 

The  difference  between  the  number  of  variations,  pro- 
duced in  the  signs  of  these  functions,  by  the  substitution  of 
p  and  of  q,  is  always  equal  to  the  number  of  real  roots  of 
the  equation  JzrO,  included  between  those  quantities;  i.  *-. 
between  p  and  q. 

a.)  When  X'  —  0,X  and  X2  have  unlike  signs  (§  382). 

But  when  X'  =  0,  X  is  alternately  positive  and  negative. 
Therefore  X2  is  alternately  negative  and  positive. 

This  principle,  which,  of  course,  supposes,  that  X=  0  has 
real  roots,  will  enable  us  better  to  understand,  how  the 
ries  of  signs  loses  a  variation  in  passing  each  real  root  of 
X=0. 

b.)   (1.)   We  may  find  simply  the  whole  number  of  real 
roots,  by  substituting  —  o>  and  -|-  o>  for  x  in  the  several 
functions.     In  this  case,  each  function  will  have  the  sigi 
of  its  first  term. 

(2.)  Moreover,  if  we  substitute  0  for  x,  the  number  of 
variations  lost  from  —  oo  to  0  will  give  the  number  of  neg- 
ative roots.;  from  0  to  -f-  oo,  the  number  of  positive  root.-. 

It  is  obvious  also,  that  the  substitution  of  0  for  x  will  re- 
duce each  function  to  its  last  term,  which  is  independent. 
of  x. 

§  387.  The  theorem  has  been  demonstrated  on  the  hy- 
pothesis, that  the  equation  contains  no  equal  roots  (§  379). 

If,  however,  we  have  an  equation  containing  equal  roots, 
we  shall  find  a  common  divisor  of  Xand  X' ;  and  a  remain- 
der, of  course,  equal  to  zero. 

If  now  we  divide  the  functions,  X,  X',  &c,  by  this  great- 
alg.  §  26 


302  THEORY  OV  EQUATIONS.  [c  888. 

est  common  divisor,  we  shall  obtain  a  new  series  of  func- 
tions, T,  T',  r2,  &c.  Now,  it  is  evident,  (1.)  that  Y=  0 
will  contain  no  equal  roots;  and  (2.)  that  the  variations  of 
sign  in  the  series  of  new  functions  will  be  the  same  as  in 
the  primitive  series. 

For,  if  the  common  divisor  be  positive,  the  signs  will  not 
be  affected  by  the  division  (§§  62.  a;  80.  b);  and,  if  it  be 
negative,  all  the  signs  will  be  changed. 

Hence,  the  theorem  is  applicable  to  equations  having 
equal  roots. 

§  388.    1.    How  many  real  roots  has  the  equation, 
X  =  a;3  —  7*4-6  =  0? 

Here  X' =  3xs  —  1 ; 

v    —  7^       o 

•r      X  -4- 

X  X'  x2  x3 

x  =  —  oc  gives         — r    -f"  —  ~f*>         ^  variations. 

xz=zO  "  -\-    —  —  -f~j         2  variations. 

x  —  -\-  go    "  +    +  +  +<         0  variation. 

Hence,  there  are  three  real  roots ;  one  negative,  and  'two 
positive. 

We  shall  find,  more  nearly,  the  values  of  the  roots  by 
substituting  different  numbers  for  x.     Thus, 


x  =■  —  4  gives 

-+:-+; 

3  variations ; 

x—  —  3     " 

0    +   -   +; 

x=—  2     " 

+   +   -   +. 

2  variations  ; 

*=— 1     " 

+ +, 

2  variations ; 

*=      1     " 

0 +; 

5B==        1.6" 

-    +    +    +, 

1  variation ; 

k"=      2     « 

0     +    +    +; 

x  '==     3     « 

+    +    +    +> 

0  variation. 

have  here  found  the  three  roots,  - 

-3,  1,  and  2  (the 

§  388.]  stdrm's  theorem.  303 

values  of  x  which  reduce  X  to  zero).     We  find  also  that  a 
variation  is  lost  in  passing  each  of  the  roots. 

2.  Find  the  number  and  situation  of  the  real  roots  of 
the  equation,  X=z  x2  -]-x  —  L  =  0. 

Here  X'  =  2x  + 1 ; 

X2=z+5. 
x  —  —  co  gives         -f-    —    -f" ,         2  variations ; 
x  =  0  "  —    4~+j  1  variation ; 

a;  =  +  co     "  +~h+5         0  variation. 

There  are,  therefore,  two  real  roots ;  one  positive,  and 
the  other  negative.     Moreover, 

x  =  —  2  gives         -J-    —    -|- ,  2  variations  ; 

x  =z  —  1"  —    —    -f~  >  1  variation  ; 

x  ■=. -f- 1     "  +    +    +>  0  variation. 

There  is,  then,  one  root  between  —  2  and  —  1 ;  and  one 
between  0  and  1. 

The  first  figure  of  the  negative  root  is  —  1 ;  and,  by  sub- 
stituting .1,  .2,  .3,  .4,  .5,  .6,  and  .7,  we  find  the  first  figure 
of  the  positive  root  to  be  .6. 

3.  How  many  real  roots  has  the  equation, 

X  =  x3  +  llx2  —  102x  + 181  =  0  ? 
Here      X'=  3x2 -\-  22x  —  102; 
X2  =  122*  —  393; 

Hence,  Ave  find  three  real  roots ;  one  negative,  and  two 
positive  situated  between  3  and  4. 

Now,  diminishing  the  roots  (§  367)  of  the  equations, 
X  =  0,  X'  =  0,  &c,  by  3,  we  find 

T'  =  3y2  +  40?/  —  9  ; 
r2z=122y-27; 


304  THEORY  OF  EQUATIONS.  [§  388. 

These  functions  show  that  the  two  positive  roots  of 
Y=  0  lie  between  .2  and  .3.  Consequently,  the  two  pos- 
itive roots  of  X  =  0  are  between  3.2  and  3.3. 

Again,  diminishing  the  roots  of  Y=  0,  T'  =  0,  &c.  by 
.2,  we  find 

Z  —  Z3  +  20.6s2  —  .88z  +  .008 ; 
Z'=  3z*  +  41.22  —  .88; 
Z2  =  122s  — 2.6; 

Hence,  the  initial  figures  of  the  two  positive  roots  of 
Z  ~  0  are  .01  and  .02.  Consequently,  the  first  three  fig- 
ures of  the  positive  roots  of  X  =  0  are  3.21  and  3.22. 

Also,  the  sum  of  the  roots  (§  355.  1)  is  — 11. 

— 11  —  3.21  —  3.22  =  — 17.4,  the  negative  root. 

4.   How  many  real  roots  has  the  equation, 

Xz=x5-\-  2x±  +  3x3  -f-  4x2  _j_  §x  _  20  —  0  ? 
Here         X'  =  5x*  +  8x3  +  9x2  +  8x  +  5  ; 

X2  =  —  7x3  —  21z2  —  42a;  •+-  255  ■> 

X3=  —  13x  +  U; 

x  =  —  co  gives     —    -f-    -f-    -f-    — ,     2  variations. 

aj=  +  oo     "         +    +    —    —    —  ?     1  variation. 

Hence  the  equation  has  one  real,  and  four  imaginary 
roots.  The  real  root  is,  of  course,  positive  (§  357.  b) ;  and 
is  found  to  be  between  1  and  2. 

a.)  "When  we  arrive  at  a  function,  as  Xm,  such  that  the 
roots  of  Xm  =  0  are  all  imaginary,  we  need  not  continue 
the  divisions. 

For  this  function  having  the  same  sign  for  all  values  of 
x  (§  358.  1),  can  never  conform  to  the  signs  of  those  be- 
yond it;  and  no  changes  of  sign  in  those  functions  can  af- 
fect the  number  of  variations  in  the  series  (§  384). 

The  coefficients  of  an  equation  of  the  second  degree, 
show  at  once,  whether  its  roots  are  imaginary  (§  216). 


k  388.]  STURM'S  THEOREM.  305 

In  respect  to  equations  of  higher  degrees,  the  question 
is  not  so  easy  of  solution.  It  can,  however,  be  determined 
by  applying  Sturm's  theorem,  as  to  an  independent  equa- 
tion. 

The  roots  of  X'  =  0,  in  the  last  example,  are  all  imagi- 
nary ;  and  X  and  X'  give  the  same  result  as  the  whole  se- 
ries of  functions. 

Note.    X'  =0  is  a  recurring  equation  (§371),  and  can  be  easi- 
ly solved  by  a  process  which  will  be  explained  hereafter. 

5.    How  many  real  roots  has  the  equation, 

X  =  x3  -\-px -\-q  z=:  0? 

Here  X'  =  3x2  +p; 

X2  zz:  —  2px  —  3q  ; 

X3  —  —  4p3  —  27q'2. 

b.)    First,  let  p  be  positive. 

Then  —  2p  will  be  negative;  and  X2  will  be  positive 
for  x  =  —  oo  ;  negative,  for  x  =  -f-  oo. 

Also,  — 4p3  will  be  negative;  and,  as  — 27q2  is  neces- 
sarily negative,  X3  will  be  negative.     Thus, 

x  =  —  co  gives     —    +    +    —  >     too  variations  ; 
x  =  -|-  co     "         +    +    —    — ,     one  variation. 
Hence,  if  p  be  positive,  the  equation  has  one  real,  and 
two  imaginary  roots. 

c.)   Again,  let  p  be  negative. 

Then  —  2p  will  be  positive ;   and  X2  will  be  negative 
for  x  =  —  co  ;  positive,  for  x  =  -J-  oo. 
Also,  — 4p3  will  be  positive;  and  when 

—  4p3>27y*,   i.  e.  when   —  4p3  —  27q2>0, 
or  (§  146.  d)  4p3  +  27«72<  0,         X,  will  be  positive. 

If  these  conditions  be  fulfilled,  we  shall  have, 
for  x  =.      co,       —    -j-    —    -f- }         three  variations ; 
x  =:  -J-  co,       — I —    — | —    — J —    — f —  ^         no  variation. 
^  Hence,  if  both  p,  and  4p3  +  27q2  be  negative,  the  equa- 
tion has  three  real  roots. 

*26 


306  THEORY  OF  EQUATIONS.  [§  389. 

d.)   Again,  suppose  X3  to  be  positive ;  i.  e. 

4p3  +  27?2<0. 
Then  4p3  <  —  27?2.  §  144  N. 

•-.(§147)    ^<-^or(|)3<-(|)2<0.     ,.P<0. 

For  —  f|j    is  negative.     Consequently,  f||)    is  nega- 
tive ;  which  cannot  be  unless  p  is  negative. 

Hence,  if      4p3  -j-  27g2<  0,      the  roots  are  all  real. 

6.    How  many  real  roots  has  the  equation, 

X=x^-\-px  +  q  —  0? 
Here  .X'=2a:+^; 

X^^-4?.    . 
First,  if  JF2  be  positive, 

x  =  —  co  gives         -f-    —    -f- ,         two  variations ; 
x  =  -\-cx>     "  -L._L.4-,         no  variation ; 

showing  two  leal  roots. 
Again,  if  X2  be  negative, 

x  =  —  co  gives         +    —    —  >         one  variation ; 
a;  __.  -|-  co     "  4~    +    —  j         one  variation ; 

showing  no  real  root. 

Consequently,  the  roots  are  real,  or  imaginary,  accord- 
ing as  p 2  —  4^  is  positive  or  negative. 

Moreover,  when  X2  is  negative  (i.  e.  p"*  —  4^<0),  we 

have  p*<fy;  or  lp2<q;  or  QfpY  <q; 

which  can  happen,  only  when  q  is  positive. 

Hence,  the  roots  are  real,  unless  q  >  (hp)  2 >0  (§  216). 

NUMERICAL  EQUATIONS. — I.  INTEGRAL  ROOTS. 

§  389.   Let  a  be  an  integral  root  of  the  equation, 
X=zxn  +  A1xn-1+A2xn-2  .  .  +^M_1a  +  ^B  =  0,  (1) 
the  coefficients  being  all  integral. 

Then  an-\-Axa"-i  .  .  -\-An.2a^+AJl.1a-\-An  =  0.  (2) 


§  389.]  INTEGRAL  ROOTS.  307 

Transposing,  and  dividing  by  a,  we  have 
£«  =  _„-!  _^ia»-2 -An_2a-A^1}     (3) 

a  whole  number. 

Hence,  An-±- a  is  a  whole  number;  and  a  is  an  integral 
factor,  or  divisor  (§  80.  d)  of  A  „. 

Consequently,  all  the  integral  roots  of  an  equation  will 
be  found  among  the  divisors  of  the  last  term.  They  will 
also,  of  course,  be  contained  between  the  superior  and  in- 
ferior limits  (§  374)  of  the  roots. 

Therefore,  we  shall  find  all  the  integral  roots  of  an  equa- 
tion, by  the  method  of  §§  349.  d,  350.,  if  we  substitute  for 
a,  successively,  the  several  factors  of  the  last  term,  which 
are  included  between  the  limits  of  the  roots. 

1.    Find  the  integral  roots  of  the  equation, 

Here,  the  limits,  found  by  §  374,  are  18  and  — 5.  It  is 
evident,  however,  that  there  can  be  no  integral  root  great- 
er than  6. 

Hence,  the  only  numbers  to  be  tried  are  6,  3,  2,  1,  —  1, 
—  2,  and  —  3. 

1  —  7  +  17  —  17  +  6     (1 

+  1—     6  +  11  —  6 


1  —  6  +  11  —     6 

(2 

+  2  —     8  +     6 

1  —  4+3             (3 

+  3—3 

1-1             (1 

+  1 

1. 

Consequently,  the  root3  are  1,  1,  2,  and  3  (§  355.  e.  2). 

2.   Find  the  integral  roots  of  the  equation, 
X  —  x3  +  x2  —  17a:  +  15  =:  0. 
If  X  is  divisible  by  x  —  a,  it  is,  evidently,  divisible  by 
a  —  x  -,  the  signs  merely  of  the  quotient  being  different, 


308 


THEORY  OF  EQUATIONS. 


[§  389. 


Therefore,  arranging  the  coefficients  according  to  the  as- 
cending powers  of  x  (§  33),  and  dividing  by  3 — x,  we  have 

15  —  17  +  1  +  1  3 

_|_     5_4_i     _|_i 


—  12 


5  —     4  —  1 

+     5  +  1 


5  + 


+ 


1 
1 
5 
1 


0 


—  1. 

Hence,  the  roots  are  3,  1,  and  — 5. 

a.)  In  this  process,  the  root,  evidently,  must  divide  the 
first  term  of  each  remainder ;  i.  e.  the  sum  of  each  term  of 
the  quotient  and  the  succeeding  coefficient. 

b.)  In  fact,  transposing  An^l  in  (3),  representing — " 
+  A  „_  1  by  B,  and  dividing  again  by  a,  we  have 


a 


B 


a 


a 


n-2 


—  Axan~" 


-™-n—3a -^n— 2> 


a  whole  number. 

In  like  manner,  continuing  to  transpose  the  coefficient  of 
a0,  and  divide  by  a,  each  quotient  will  be  a  whole  number ; 
and  the  last  quotient  will  be  the  coefficient  of  xn  with  its 
sign  changed. 

3.    Find  the  integral  roots  of  the  equation, 

x4  —  21x-  +  14k  + 120  =  0. 

Here  +7  and  — 7  are  limits.  Moreover,  only  two  of  the  roots  can 
be  negative,  and  two,  positive  (§361).  Hence,  having  found  two 
positive  roots,  we  need  try  no  more  positive  divisors. 


6 

+  1 


120  +  14  —  27 
+  20 
+  34. 


0  +  1 


34-^-6  not  being  a  whole  number,  6  is  not  a  root. 

The  roots  are  4,  3,  —  2,  and  —  5. 


§  390.]  INCOMMENSURABLE  ROOTS.  309 

c.)  If  the  equation  is  not  in  the  common  form  (i.  e.  with 
integral  coefficients,  the  first  being  unity),  it  should  be  re- 
duced (§  369),  and  the  method  applied  to  the  reduced  equa- 
ion. 

4.    Find  the  roots  of  the  equation, 

3x3  —  x2  —  3x  +  9  =  0. 
Having  found  the  values  of  y  from  the  transformed 
equation,  we  shall  have  x  =  \y. 

II.  INCOMMENSURABLE  ROOTS. 

§  390.    Find  the  roots  of  the  equation, 
X=x-  +  5x  —  5  =  0. 
Applying  Sturm's  theorem,  we  find 
X  =  x2  +  5x  —  5  ; 

X'  =  2x  +  5  ; 

x2  =  +. 

Hence  there  is  a  positive  root  between  .8  and  .9,  and  a 
negative  root,  between  —  5  and  —  6. 

If,  now,  we  diminish  (§  367)  the  roots  of  X  —  0  by  .8, 
one  root  of  the  transformed  equation, 

Z=y2  +  6%  -.36  =  0, 

will  be  between  0  and  .1. 

Applying  Sturm's  theorem  again,  we  find 
Y=y2+G.6y  —  .36  =  0; 
r'  =  ^/  +  3.3; 

r2  =  +  . 

Hence,  there  is  a  root  between  .05  and  .06 ;  and,  conse- 
quently, the  root  of  1=  0  is  between  .85  and  .86. 

Again,  diminishing  the  roots  of  T=  0  by  .05,  one  root 
of  the  transformed  equation, 

Z  —  z2  +  Q.7z  —  .0275  =  0, 
will  be  between  0  and  .01 ;  and  will  be  found  by  the  the- 
orem to  be  between  .001  and  .005, 


310  THEORY  OF  EQUATIONS.  [§391,392. 

Hence,  the  root  of  X  =  0  is  between  .854  and  .855. 

Note.  We  might,  in  the  same  way,  find  any  number  of  figures 
of  the  root.  But  the  process  would  be  tedious.  The  nature  of  the 
roots,  however,  of  the  equations,  F=0  and  Z-0,  will  suggest  a 
more  convenient  method  of  determining  the  successive  figures,  as  ap- 
pears in  the  following  sections. 

§  391.   We  know  that  the  root  of  the  equation, 

?2  +  6%-.36  =  0,  (1) 

is  less  than  .1 ;  i.  e.  we  have  y  <  .1,  and,  of  course,  y2<  .01. 
Hence  it  is  evident,  that  the  equation, 

6%  +  .36r=0,  (2) 

will  furnish  a  near  approximation  to  the  true  value  of  y. 

or? 

\    In  fact,  we  have,  from  (1),      y  = 


y+6.6' 

in  which  the  first  significant  figure  will  be  the  same,  wheth- 
er we  take  y  :=  0,  or  .09,  as  will  be  seen  by  dividing  .36, 
successively,  by  G.6,  and  by  6.69. 

The  same  reasoning  will  apply,  with  still  greater  force, 
to  the  first  figure  of  the  root  of  Z=  0. 

Hence,  we  may  find,  in  each  instance,  approximately, 
the  next  figure  of  the  root  by  dividing  the  coefficient  of  y° 
and  z°  by  the  coefficient  of  y1  and  z1. 

The  operation,  then,  will  stand  thus  ; 

1         _}-  5.8  —5         (.8541 

.8  -f  4.64 

—  .36 
-f  .3325 

—  .0275 
+  .026816 

—  .000684 
+  .00067081 

6.7082  —  .00001319 

§  392.   To  explain  this  method  of  solution  in  a  more 
general  form,  let  a  root  of  the  equation, 


6.65 

5 

6.704 

4 

6.7081 

1 

§393.1  INCOMMENSURABLE  KOOTS.  ii 

X^^  +  A^'-i  +  Asx"--*  ....  +  A^vc-{-An^0, 
be  x  =  x'-\-y;  x1  being  the  part  of  the  root  already  found, 
and  y  representing  the  remaining  figures,  and  y  being,  of 
course,  very  small  compared  with  x'  (§  174.  N.  1). 

Then  diminishing  the  roots  of  X  —3  0  by  x1,  we  have 
frr^+^f-1  ....   +Bn_0y*  +  Bn_ly  +  £n=n. 

But,  y  being  very  small,  its  powers  above  the  first  may, 
for  the  moment,  be  neglected  ;  and  we  shall  have,  nearly, 
Bn-xy  +  Ba=.0i 

or,  also  approximately,         y  —  —  ~  "  . 

The  correctness  of  the  result  will  be  verified  by  intro- 
ducing into  the  transformed  equation  the  figure  so  found. 

Representing  the  figure  so  found  by  y',  we  shall  have 
y  =  y~|-  z  ;  and  finding  an  equation,  whose  roots  are  less 
than  those  of  Y=  0  by  yr,  we  shall,  in  like  manner,  find 
another  figure  of  the  root ;  and  so  on. 

Hence,  for  finding  a  root  of  an  equation  of  any  degree 
whatever,  we  have  the  following 

RULE. 

§  393.  1.  Find  by  Sturm's  theorem,  or  by  trial,  the 
first  figure,  or  the  integral  part,  of  the  root. 

2.  Transform  the  equation  into  another,  whose  roots 
shall  be  less  than  those  of  the  given  equation  by  the 
part  of  the  root  already  found. 

3.  With  the  last  coefficient  of  the  transformed  equa- 
tion for  a  dividend,  and  the  last  but  one  for  a  trial 
divisor,  find  the  next  figure  of  the  root ;  and  verify  it 
by  substitution  in  the  transformed  equation  (§  350). 

4.  Diminish  the  roots  of  the  transformed  equation 
by  the  figure  just  found,  divide  as  before  for  the  next 
figure ;  and  so  on,  as  far  as  is  necessary. 


312  THEORY  OF  EQUATIONS.  [§394. 

a.)  The  method  is  applicable  to  both  positive  and  nega- 
tive roots ;  each  figure  of  a  negative  root  being  treated,  in 
multiplying,  as  a  negative  quantity. 

h.)  A  negative  root  is,  however,  more  conveniently  found 
by  changing  the  signs  of  the  alternate  terms,  and  finding 
the  corresponding  positive  root  (§359). 

394.    1.    Find  the  roots  of  the  equation. 

X  =  x*  +  10x°  +  ox  —  260  =  0. 
Here,      X'  =  3x  2  -f  20a:  +  5  ; 
X2=  17*  +  239; 

x  c=  —  oo  gives  two  variations ;  x=--\-  ao,  one.  Hence, 
there  is  but  one  real  root ;  positive,  of  course  (§  357.  b). 

We  find,  moreover,  that  the  first  figure  of  the  positive 
root  is  4. 


+  10 

+  5 

—  260     (4.1179 

4 

56 

+  244 

14 

61 

-16 

4 

72 

'+  13.521 

18 

,133 

,—  2.479 

4 

2.21 

1.376531 

,22.1 

135.21 

—  1.102469 

1 

2.22 

.966221613 

22.2 

,137.43 

,—  .136247387 

1 

.2231 

.124396356339 

,22.31 

137.6531 

,—  .011851030661 

1 

.2232 

,137.8763 

22,32 

1 

.155359 

,22.337 

138.031659 

7 

.156408 

22.344 

,138.198067 

7 

.201167] 

L 

_    ,22.3519 

138.21817371 

§  394.]  INCOMMENSURABLE  ROOTS.  313 

The  coefficients  of  the  successive  transformed  equations 

are  marked  with  commas,  the  first  coefficient  in  each  being 

the  same  as  in  the  primitive  equation.     Thus,  we  shall 

have 

r  =  yz  +  22y2  +  133y  —  16  =  0; 

Z  =  z"  +  22.3^2  4- 137.432  —  2.479  =  0  ;  and  so  on. 

Note.  It  will  be  observed,  that  the  .1  added  to  22,  does  not  term 
a  part  of  the  coefficient  of  ?/2,  but  was  added  to  that  coefficient  in 
forming  the  next.  A  similar  remark  applies,  of  course,  to  the  subse- 
quent coefficients;  and  to  the  example  of  §  391,  where  .8  is  most  con- 
veniently added  to  5,  by  being  written  after  it. 

a.)  The  coefficients  of  the  two  last  terms  (Bn_l,  and  Bn, 
[§  392])  in  each  of  the  transformed  equations  have  unlike 
signs.     This  is  as  it  should  be,  in  finding  a  positive  root. 

For,  suppose  that  the  least  real  root  of  X  =  0  is  posi- 
tive ;  and  represent  the  part  already  found  by  x1 . 

Then  Bn  and  -S„_r  are  what  Xand  X  become,  when  a/ 
is  substituted  for  x.  Therefore,  x1  being  less  than  the  least 
real  root,  Bn  and  Bn_x  (i.  e.  /(«')  and  /'{x1))  must  have 
unlike  signs  (§§373.  b;  385). 

b.)  Similar  reasoning  will  apply  to  any  other  positive 
root,  provided  xf  differs  from  that  root  less  than  the  next 
inferior  root  of  X'  =  0  does  (§  385.  a).     See  g,  h,  below. 

c.)  In  approximating  to  a  negative  root  (§394.  a),  xf  is 
greater  than  the  root ;  and,  of  course,  if  it  is  less  than  the 
next  greater  root  of  X'  =  0,  Bn  and  Bn_x  (i.  e.  /{xf)  and 
/'(a/)),  must  have  like  signs. 

d.)  If,  having  found  the  root,  4.1179,  we  divide  X  by 
x  —  4.1179,  we  shall  have  an  equation  of  the  second  degree, 
from  which  we  may  find  the  remaining  roots  (§  353.  c). 

e.)  Otherwise  ;  we  know  that  the  coefficients  of  x2  and 
x°  in  the  given  equation  are  respectively  the  sum,  and  pro- 
duct of  the  three  roots  with  their  signs  changed.  Also,  the 
coefficients  of  xl  and  x°  in  the  depressed  equation  will  be 
the  sum  and  product  of  the  two  remaining  roots  with  their 
signs  changed  (§  355.  1,  4). 
alg.  27 


'jl4  theory  of  equations.  [§394. 

Hence,  if  we  diminish  the  coefficient  of  x2,  and  divide 
the  coefficient  of  x°,  in  the  given  equation,  by  the  root 
found  taken  with  a  contrary  sign,  we  shall  have  the  coeffic* 
ients  of  a;1  and  x°  in  the  depressed  equation.     Thus, 

or  x2  +  14.1179  x  +  63.1365  =  0, 

will  give  the  remaining  roots  of  the  equation,  which,  are, 

evidently,  imaginary  (§  388.  6). 

f.)  "When  the  roots  are  all  real,  it  is  frequently  quite  as 
convenient  to  find  a  second  root  from  the  given  equation, 
in  the  same  manner  as  the  first ;  and  then  find  the  third  by 
adding  the  two  roots  found  to  the  coefficient  of  x~,  and 
changing  the  sign  of  the  result  (§§  355.  1 ;  388.  3). 

3.    Find  the  roots  of  the  equation, 

X  =  x3  —  7x  +  7  =  0. 
Here  X' =  3x*  —  7; 

X2  =  2x  —  3  ; 

Hence,  there  are  three  real  roots ;  one  between  —  3  and 
—  4,  and  two  between  1  and  2.  Also,  the  first  two  figures 
of  the  roots  are  — 3.  0,    1.3,   and  1.6. 

To  find  the  greatest  root,  proceed  thus. 

x 


0 

—  7 

+  7     (1.69202147 

1 

1 

—  6 

~L 

—  6 

,+  1 

1 

2 

—  1.104 

~2 

-4 

,—  .104 

1 

2.16 

.100809 

,3.6 

—  1.84 

,—  .003191 

.6 

2.52 

4.2  ,+  .68 

Find  the  other  figures  of  the  root.     Also  find  the  other  root. 

g.)    There  are  here  two  roots  of  X  =.  0,  and  only  one 

of  X'  =  0  (viz.  1.528),  greater  than  1. 


§  394.] 


INCOMMENSURABLE  ROOTS. 


315 


Consequently,  the  substitution  of  1  for  x  renders  X  pos- 
itive and  X'  negative  (§  372) ;  giving 

Bn=f<P)  =  l,    andBn.1=f(x')  =  -A. 

Again,  1.6  being  greater  than  the  greatest  root  of  X'  =  0, 
and  less  than  that  of  X=  0,  renders  X  negative  and  X1 
positive  (§  372)  ;  giving 

Bn  =  f(x>)  =-.104,    and  BH_l  =  f(x')  =  +  .68. 

If  we  had  substituted  1.5,  Bn_1  would  have  remained 
negative;  because  1.5  is  less  than  the  greatest  root  of 
X'  =  0. 

Hence,  if  the  sign  of  Bn  changes,  that  of  Bn^1  should 
change  also.     See  a,  above. 

h.)  It  may,  however,  not  change  at  the  same  figure  of 
the  root,  for  that  figure  may  be  common  to  the  next  great- 
er root  of  X=  0  and  of  X'  =  0.  This  occurs  in  the  great- 
est root  of  the  following  equation.     See  4,  below. 

i.)  To  find  the  negative  root,  we  change  the  signs  of  the 
alternate  terms  (§  393.  b). 

1—0  —7  —7     (3.048917 


3 
3 


o 
O 


2 

18 
,20 


_6 
,— 1 


.814464 


.3616 


,9.04 
4 

9.08 
4 


20.3616 

.3632 

,20.7248 


.0730 


,9. 

128 

8 

9. 

136 

8 

,9. 

1449 

9 

T. 

1458 

20.7978 
.0730 


,20.8709 
82 


24 
24 

88 
12 
3041 


,—  .185536 
.166382 

,—  .019153 
.018791 

,—  .000362 
208 

592 

408 

228169 

179831 

873763 

,—  .000153 

146 

,—  .000007 

306068 
211615 
094453 

20.879 
8 


14241 
23122 


20|8|8]737363         x  —  —3.048  917. 


316  THEORY  OP  EQUATIONS.  [§394, 

k)  "We  should  evidently  have  obtained  the  same  result, 
as  far  as  we  have  carried  the  approximation,  and  with 
mnch  less  labor,  if  we  had  neglected  all  the  figures  on  the 
right  of  the  vertical  lines  in  the  several  columns. 

4.  Find  the  roots  of  the  equation, 

X  =  x3  + 1 1x2  _.  i02x  +  180  =  0  (§  388.  3). 

The  roots  are  3.229  52,  3.213  127  7,  and  — 17.442  648  96. 
The  greatest  root  of  X'  =  0  is  3.2213.     Consequently,  3.22  substi- 
tuted for  x,  will  render  both  X  and  X'  negative.     But  3.229  will 
render  X  negative,  and  X  positive.     See  h  above. 

5.  Find  the  roots  of  the  equation, 

8x3  —  fry  — 1  =  0. 
It  is  not  necessary  for  the  application  of  Sturm's  theo- 
rem, or  of  this  method  of  approximation,  to  reduce  the 
equation  as  in  §  389.  c. 

We  shall  find,  that  there  are  three  real  roots ;  one  posi- 
tive, and  two  negative ;  and  that  their  initial  figures  are 
.9,  —  .1,  and  —  .7. 

The  equation  may  be  put  under  this  form, 

x&  _  |aj—  £  =  x3  —  .75a:  —  .125  =  0. 
To  find  the  negative  root,  proceed  as  follows. 
1  0.7  —.75  +.125     (.76  &c. 

.7  _^49  —.182 

IX  =^26  ,—  .057 

.7  .98  .050976 

,2T6  ,+  772  -  .006024 

.1296 
.8490 
The  roots  are  —  .760  04,  —.1737,  and  .9397. 

6.  Find  the  real  root  of  the  equation  a:3  —  2  =  0  ;  i.  e. 
find  the  cube  root  of  2.  Ans.  1-259  921. 

7.  Find  the  roots  of  the  equation  a;2  —  2  =  0  ;   i.  e. 
find  the  square  root  of  2.  Ans.  ±1.414  213  6. 

Note.    It  will  be  observed,  that  the  solution  of  the  third  and 
fourth  examples  i3  equivalent  to  the  processes  of  §§  174,  179.    The 


§  394.]  INCOMMENSURABLE  ROOTS.  317 

method  is,  obviously,  equally  applicable  to  the  extraction  of  roots  of 
large  numbers.  The  trial  divisor,  however,  approximates,  of  course, 
most  closely  to  the  complete  divisor,  when  the  part  of  the  root  not 
yet  found  is  very  small. 

8.  What  is  the  cube  root  of  3  442  951  ? 

Ans.  151. 

9.  Find  the  roots  of  the  equation, 

x*  —  12x*-\-12x  —  3  =  0. 


0 

—  12 

+  12 

—  3  (2.8  &c 

2 

4 

—  16 

—  8 

"2 

—  8 

—  4 

-11 

2 

8 

0 

8.9856 

4 

0 

,-4 

,—  2.0144 

2 

12 

15.232 

6 

,12 

11.232 

2 

7.04 

^8.8 

19.04 

Am.  2.858  083,  .606  018,  .443  276  9,  and  —3.907  378. 

Continue  the  operation,  and  find  the  other  roots.  The  work  may 
be  Teatly  abridged  by  rejecting  all  but  one  decimal  figure  in  the  col- 
umn of  x3,  two  or  three  in  the  column  of  x2,  three,  four  or  five  io 
that  of  xl,  and  four,  five,  six  or  seven  in  that  of  xo. 

10.  Find  the  roots  of  the  equation, 

x3  —  2x  —  5  =  0. 
Ans.  2.094  55  ;   the  other  roots  are  imaginary. 

11.  Find  the  roots  of  the  equation, 

x±  —  x2-\-  2x  —  1  =  0. 
Ans.  0.618,  and  — 1.618  ;  the  others,  imaginary. 

12.  Find  a  root  of  the  equation, 

X5  _|_  2a:*  +  3x3  +  4x2  +  5x  —  54321  =  0. 

Ans.  x  =  8.414  454  7. 

13.  What  is  the  fifth  root  of  2  ?  Ans.  1.148  699. 

Note.    This  method  of  finding  the  real  roots  of  any  equation,  if 
incommensurable,   approximately,    if   commensurable,    exactly,    is 

sometimes  called  Horner's  method. 

*27 


318  -      THEORY  OP  EQUATIONS.  [§  395. 

RECURRING,  OR  RECIPROCAL  EQUATIONS. 

§  395.    The  general  form  of  a  recurring,  or  reciprocal 
(§  371)  equation  of  an  odd  degree,  is,  obviously, 
x2»+i+^1a;2n  +  A2x2n~i  . .  ±  A2x2±  Axx  ±  1  =  0  ;  (1) 
in  which  the  like  coefficients  belong,  one  to  an  even,  and 
the  other  to  an  odd  power  of  x  throughout. 

Now,  if  the  corresponding  coefficients  have  like  signs, 
the  substitution  of  —  1,  and,  if  they  have  unlike  signs,  the 
substitution  of  -f- 1,  for  x,  will  render  the  corresponding 
terms  numerically  equal  with  contrary  signs ;  and  will, 
therefore,  reduce  the  first  member  to  0.     Hence, 

One  of  the  roots  of  a  recurring  equation  of  an  odd  de- 
gree is  —  1,  or  -f- 1,  according  as  the  corresponding  coeffi- 
cients have  like  or  unlike  signs. 

a.)  Again,  the  equation  may  be  written  thus, 
(x*"+i±l)-\-Alz(x*»-1±l)+A<ix*(x2n-z±l)  .  .  =  0;(2) 
in  which  x  =  —  1,  if  we  take  the  upper  signs,  and  x  =z 
-L-l,  if  we  take  the  lower  signs,  will  render  each  of  the 
quantities  enclosed  in  parenthesis  equal  to  zero. 

b.)   Let  2n  -\- 1  =  5.     Then  the  equation  becomes 
xsJrAlx*  +  A2xZ±A2x2±A1x±l  =  0;         (3) 
or     (x&±l)  +  A1x(x?&l)±Azx*(x±l)  =  0.        (4) 

Now,  if  we  divide  either  the  first  member  of  (3)  or  each 
term  of  (4)  by  x  ±  1  (§§  98,  96),  taking  always  the  upper 
signs  together,  and  the  lower  signs  together,  we  shall  have 


x4  q:   1 


x2qp    1 

-Mi 


£-1-1=0;    (5) 


x3+  1 

+  4i 

evidently  an  equation  of  an  even  degree  (the  2«tn),  whose 

coefficients  at  equal  distances  from  the  extremes  are  equal 
(i.  e.  are  numerically  equal  and  have  like  signs).  It  is, 
therefore,  a  recurring  equation  (§  370.  b). 

The  same  reasoning  will,  obviously,  apply  to  any  similar 
equation  as  well  as  to  that  of  the  fifth  degree. 


§  396-7.]    RECURRING,  OR  RECIPROCAL  EQUATIONS.    319 

§  396.    The  general  form  of  a  recurring  or  reciprocal 
equation  of  an  even  degree,  in  which  the  like  coefficients 
have  unlike  signs  and  the  middle  term  is  wanting  (§  370. 
c),  is,  obviously, 
x2n+2_|_^i;c2.H-i   #  .  -\-0xn  —  .  .  —Axx—l  —  0.    (6) 

Arranging  (§  34.  c)  according  to  the  coefficients,  we  have 
(x5"+-— 1)+J1x(x2n—  l)+A2zi(x«-n-2—l)  .  .  =  0  ;  (7) 
each  term  of  which  is,  evidently,  divisible  by  x2 — 1  (§  96), 
i.  e.  by  (x+l)(x—l)  [§  93].     Hence, 

A  recurring  equation  of  an  even  degree,  in  which  the 
middle  term  is  wanting  and  the  corresponding  coefficients 
have  unlike  signs,  has  its  first  member  divisible  by  x2 — 1 ; 
and,  of  course,  has  the  two  roots,  —  1  and  -f-  1. 

a.)  Let  2n-\-2  =  6.     Then  the  equation  becomes 
x*-{-A1x5-\-A2x'i  —  A2x2  —  A1x  —  l  =  0;       (8) 
or     f(a;°-  1)  +  Axx{x±  —  l)+A2x°-(x*  —  1)  =  0.     (9) 

Now  if  we  divide  either  the  first  member  of  (8)  or  each 
term  of  (9)  by  x2 — 1,  we  shall  have 

x*  +  Axx^-\-A2     x2  +Alx  +  1=0;        (10) 

+  1 
a  recurring  equation  of  an  even  degree,  whose  like  coeffi- 
cients have  like  signs,  as  in  §  395.  b. 

b.)  Otherwise ;  the  roots  of  the  depressed  equations,  (5) 
and  (10),  are  the  remaining  roots  of  the  primitive  equations 
(3)  and  (8)  ;  and  one  half  of  them  are,  therefore,  the  re- 
ciprocals of  the  other  half. 

§  397.  The  general  form  of  a  recurring  equation  of  an 
even  degree,  in  which  the  like  coefficients  have  like  signs,  is 
x2n -\- Axx2n~^  .  .  +Anxn  .  .  +  .41a;  +  l=:0.         (11) 

Dividing  by  xn,  we  have 

x  -\-  A.^x  ~     .  .  .  -\-A.n^]X  -\-  A.n  -J-  ^lu_1—  .  .  . 


x 


+  ^i=i+=--^  (12> 


or 


i  -vt —  A      ■     2*t 


320  THEORY  OF  EQUATIONS.  [§  398. 

*«+^+A(xn-1+^) '  •  +A»-i(x+l)+A»  =  0' (13) 

Now  put  x-\-  -  =  z.  (a) 

JO 

Then,  squaring  and  transposing, 

x*+±;  =  z*-2.  (b) 

JO 

Multiplying  (b)  by  x  -\-  -  =  z, 

x34-^--\-x-\--  —  z^—2z ;  orx3+-!-  =  zS—Sz.  (c) 

X3  X  xd 

Or,  in  general,  since 

we  have,  by  transposing, 
^'+^T=(^+p)(-+i)-(--'+-if)-      (4 

Thus,  making  to  =  3, 

from  (a),  (3),  and  (c), 
x*+  i  =  (z^_  3*)s  -  (*»-  2)  =  z*-  4*2+  2.  (/) 

Substituting  these  values  of  x-\-x~1,  x2-\-x~2,  &c.  in 
(13),  we  shall  have  an  equation  of  the  nth  degree  in  z;  i. 
e.  of  half  the  degree  of  the  primitive  equation,  (11).    Hence, 

§  398.  A  recurring  equation  of  an  even  degree,  in  which 
the  like  coefficients  have  like  signs,  can  always  be  reduced 
to  an  equation  of  half  that  degree. 

a.)   Hence  (^§  395,  396), 

Cor.  A  recurring  equation  of  an  odd  degree  (2»  -f- 1), 
or  one  of  an  even  degree  (2n-\-2)  whose  middle  term  is 
■wanting  and  whose  like  coefficients  have  unlike  signs,  can 
always  be  reduced  to  an  equation  of  the  «th  degree. 

b.)  The  solution  of  the  equation  of  the  nth.  degree  gives 


§  398.]     RECURRING,  OR  RECIPROCAL  EQUATIONS.        321 

the  values  of  *;   and  the  values  of  a?  may  be  found  from 
the  equation, 

x-J--  =  z;    i.  e.  a?2 — zx  = — 1. 
x 

1.   Find  the  roots  of  the  equation. 

X5_  na;4_|_  I7a?3+I7a?2—  \\x  + 1  =  0. 

One  root  is  —1  (§  395).     Therefore,  dividing  by  *  + 1 
(§  348,  350),  we  have 

x*—  12a?3-|-29a?2—  12a? +1  =  0. 

Dividing  by  a?2,  (*2+^)-  12(^  +  ^)+  29  =  °- 

Substituting, 

z2_  2  —  12c  +  29  =  0 ;  or  s*— 12*  _|_  27  =  0. 

z  =  a?  4--  =  9,  or  3. 
a? 

3f2=»j     a?2—  9a?  =  —  1,  and  x  =  i(9±y77)  ; 

if  2  =  3,     a?3— 3a?  =  — 1,  and  a?  =  £(  Si^/5). 

Therefore,  the  five  roots  are 

9  +y77      9  — y77      3  +y5  3  — y5 

-1, g ,  g       '    — 2"— 'and— Y      ; 

or,  rendering  the  numerators  of  the  third  and  fifth  roots  ra- 
tional (§  187), 

9-K/77  2  3+^5    aQd_2__. 

_1'  2       '     9+^77'  2      'ana3+V5' 

the  third  root  being  the  reciprocal  of  the  second,  and  the 
fifth,  of  the  fourth  (§  120.  d). 

2.    Find  the  roots  of  the  equation, 

4a?6—  24a?5-f  57a?4—  73a?3+  57a?2—  24a?  -f  4  =  0. 
The  reduced  equation  is 

4z3—  24z2+  45s  —  25  =  0, 
whose  roots  are  1,  §  and  f. 

Hence,  the  roots  of  the  given  equation  are 

2,  h  2,  1,  1~i^,  ™*  l=^F1- 


322  THEORY  OF  EQUATIONS.  [§  399,  400. 

3.  Solve  the  equation, 

5**4-  8z3+  9^2+  8x  +  5  =  0  (§  388.  N.). 

lAns.  5z2-f8z  — 1  =0. 
x  —  .05825  ±^/(— .9966), 
x  =  —.85825  ±y(—  2634). 

4.  Solve  the  equation, 

a;«— 6|x5-f-llfx4—  llfa;2+6|a;5—  1  (§398.  a). 
The  roots  are  1,  —  1,  2,  £,  4,  and  J. 

BINOMIAL  EQUATIONS. 

§  399.   Equations  of  the  form, 

y»±A=zQ,  (1) 

containing  but  two  terms,  are  called  binomial  equations. 

Suppose  An  =.  a,  i.  e.  A  =  a". 

Then  we  have  yn±.an=  0. 

Putting  y  =  aa;,  anx"±  a"  =  0 ; 

or  xn±  1  =  0.                                  (2) 

§  400.   I.   Let  n  be  an  odd  number,  2m-\-\. 

Then  x2m+i±l  =  0,  (3) 

being  a  recurring  equation  of  an  odd  degree  (§  395),  has 
one  reed  root  equal  to  —  1,  or  -j- 1,  according  as  the  last 
term  is  positive  or  negative. 

1.   Let  the  equation  be       ar2"»+i  —  1  =  0.  (4) 

Then  -j-1  is  a  root ;  and  dividing  by  x — 1,  we  have  (§  96) 

a;2m_^a,2m-l_|_x2m-2     .    .    .    -L.  X*  _L.  a;  -f-  1   =  0,  (5) 

which  can  be  reduced  to  an  equation  of  the  with  degree 
(§  398). 

Moreover,  (4)  has  ?io  o^er  rea?  root.  For,  if  x  be  neg- 
ative, x*™+1  will  be  negative  (§  151.  c)  ;  and,  if  a;  be  pos- 
itive and  different  from  ljX2"^1  evidently  cannot  be  equal 
to  1. 


§  401.]  BINOMIAL  EQUATIONS.  323 

Consequently,  all  the  roots  of  (5)  are  imaginary, 
a.)   This  is  evident,  also,  from  the  number  (2m)  of  con- 
secutive terms  wanting  in  (4).     See  §  364.  2. 

2.   The  equation,         x2ni+l  + 1  =  0,  (6) 

has  (§  395)  one  real  root  equal  to  —  1 ;  and,  reasoning  as 
above,  it  is  evident,  that  it  can  have  no  other  real  root. 

If  we  divide  by  x-\-l  (§  98),  we  shall  have  the  equation 

containing  the  remaining  roots,  which  can  be  reduced  by 
§397. 

b.)  Also,  the  roots  of  x-mJrl  -f- 1  =  0  are  the  same  as 
those  of  x2m+l] —  1  =  0,  taken  with  contrary  signs  (§  359). 

§  401.   II.   Again,  let  n  be  an  even  number,  2m. 

1.  Then  x2m  —  1  =  0  (8) 
has  two  real  roots,  -j- 1  and  —  1  (§396). 

It  has  also  no  other  real  roots.  For,  if  we  divide  by 
x3  —  1,  we  have 

a.2m-2_J_a.2m-4 _J_  X<1  _|_  1  _  Q  .  (0) 

in  which  the  powers  of  x  being  all  even  (§  151.  c),  any  real 
value  of  x,  whether  positive  or  negative,  will  render  the 
first  member  positive  (§  358.  3),  i.  e.  >  0. 

This  equation  can  be  reduced  also  to  one  of  the  (m — l)th 
degree  (§  398). 

a.)   Moreover,  we  have 

x"m—  1  =(xm  —  l)(xm-f-l  =  0. 
xm  —  1  =  0,  and  xm+  1  =  0. 

2.  All  the  roots  of  the  equation, 

x"m-\-l  =  0,  (10) 

i.  e.  x2"'  r=  —  1,  are  imaginary  (§  22.  2). 

This  equation  can  be  reduced  to  one  of  the  mth  degree 
(§398). 

b.)  In  each  of  the  equations,  (8)  and  (10),  there  is  a  de- 
ficiency of  an  odd  number  (2m —  1)  of  consecutive  terms- 


324  THEOBY  OF  EQUATIONS.  [§  402,  403. 

Consequently  (10)  must  contain   at  least  2m   imaginary 
roots;  and  (8),  at  least  2m —  2  (§  364.  1). 

§  402.  Let  the  real  roots  be  suppressed  from  the  equa- 
tion, x"^  1  =  0;  and  let  the  equation  in  z,  Z:=  0,  be  found 
(§397). 

Let  also  one  of  the  imaginary  values  of  x  be  a-\-by — 1. 
Then  we  shall  have 

But  (§§187,  162) 

1  a  —  by— 1  a  —  by—  1 


a  +  by—  1       (a  +  by—  1)  («  —  by—  1)  a2+6a 

Moreover,  if  a-\-by — 1  be  a   root  of  the  equation, 
xn  q:  1  =  0,  a  —  b  y —  1  must  be  a  root  also  (§  357). 

Hence  we  shall  have 
(a-{-by—l)n  =  ±l,  and  (a  —  by—l)n—  ±1. 
.-.     [(a  +  by-l){a  —  by-l)Y  =  (a*-+b2)nz=l. 

And  since  a2-\-b-  is  a  positive  quantity,  we  have 

— -r-j-- — ;  —a—  b\y—  1 ; 
a-\-by — 1 

and  z  =  a  -f-  b  y—  l-\-a  —  b  y—  1  =  2a. 

Hence,  all  the  roots  of  the  equation,  Z  —  0,  are  real. 

§  403.   Let  a  be  one  of  the  imaginary  roots  of  the  equa- 
tion, xn  —  1  =  0. 

Then  we  have     an  =  1 ;  an-n  =  1 ;  a3n  =  1 ;  &c. 
also  «-"  =  1 ;  a~2n  =  1 ;  ar*n  =  1 ;  &c.     Hence, 

If  «  be  an  imaginary  root  of  the  equation  xn  —  1  =0, 
then  will  any  integral  power  of  a  be  a  root  also. 

a.)  As  the  equation  can  have  but  n  roots,  many  of  these 
powers  of  a  must  be  equal  to  one  another. 

Thus,  the  imaginary  roots  of  a;4 —  1  —  0  are  +  */ —  * 
and  —  y—  1.    Now  we  have  ($  162) 


§  404,  405.]  BINOMIAL  EQUATIONS.  325 

-(■v/-l)2  =  -l;    (V-iy  =  -V-l;    (v>-l)*  =  l; 
(y_l)5=y_1;    (y_i)6=_1;    (y_l)7=_y_1; 

6.)  It  must  be  understood,  however,  that  these  are  only 
different  ways  of  expressing  the  same  roots.  The  equa- 
tion, xn^:l  =  0,-has  no  equal  roots ;  since  its  derived  equa- 
tion nxn~x  —  0  has  no  common  measure  with  it  (§  378./). 

§  404.    Let  a  be  an  imaginary  root  of  the  equation, 

afrf-1  =  0. 
Then  we  have 

ott=  —  1;    (aB)3==(as)B  — —  1;    (a")5  =  (a5)"  =  —  1; 
also     (a")-3  =  («-3)',  =  — 1;  (a2"1-^)"  =  —  1.     Hence, 

If  a  be  an  imaginary  root  of  the  equation,  xn  -f-  1  ==  0, 
then  will  any  odd  integral  power  of  a  be  a  root  also. 

Thus,  the  imaginary  roots  of  x~-\-  1  =  0  are  -\-»/ —  1 
and  — «/ —  1 ;  and  all  the  odd  integral  powers  of  either  of 
these  roots  are  also  roots  (§  403.  a). 

§  405.    Find  the  roots  of  the  following  equations ; 

1.     £3—1  —  0. 

Ans.  1, - ,  and . 

2.  x*—  1  =  0.         Ans.  1,  —  1,  y—  1,  and  —  y—  1. 

3.  *6_i  —  0;  i.  e.  (x3—l)(x3-\-l)  =  0. 

.  i±y—  3        _  liy—  3 

Ans.  1,  —  1, ,  and 


2         ' 2 

Note.  The  roots  of  the  equations,  a;  2 — l  =  0,  a:  3 — 1  =0,  &c, 
are  sometimes  called  the  roots  of  unity.  It  is  evident  (§§151.  a; 
152.  a),  that  the  roots  of  any  other  number,  of  any  degree,  may  ba 
found  by  multiplying  one  of  them,  most  conveniently,  the  arithmeti- 
cal root,  by  the  several  roots  of  unity  of  the  same  degree. 

ALG.  28 


CHAPTER  XVII. 


CONTINUED  FRACTIONS. 


§  406.  A  continued  fraction  is  one  whose  nu- 
merator is  a  whole  number,  and  whose  denominator 
is  a  whole  number  plus  a  fraction,  which  also  has  a 
ivhole  number  for  its  numerator,  ana  for  its  denom- 
inator a  whole  number  plus  a  fraction;  and  so  on. 

We  shall  consider  only  those,  in  which  each  of  the  nu- 
merators is  unity,  and  the  partial  denominators  (a,  below) 
are  all  positive.     Thus, 

1  (1)         1  (2) 


1' 


ai 


1  4  +  &c.  J   '  «3+&c. 

are  continued  fractions. 

a.)  The  integral  parts  of  the  denominators  are  some- 
times called  partial  denominators,  or  partial  quotients; 

and  the  fractions,  \,  -^,  &c,  — ,  — ,  &c,  are  called  partial. 

or  integral  fractions. 

§  407.  If,  in  (1)  above,  we  neglect  all  but  the  first  par- 
tial fraction,  the  denominator  2  will  be  less  than  the  true 
denominator;  and,  of  course,  |is  greater  than  the  true  val- 
ue of  the  continued  fraction. 

Again,  suppose  we  neglect  all  but  two  partial  fractions. 
Then,  the  partial  denominator,  3,  being  too  small,  the  par- 


§  408.]     APPROXIMATE  VALUES,  OR  CONVERGENTS.      327 

tial  fraction,  £,  is  too  great ;  and,  consequently,  2^  being 
greater  than  the  true  denominator,  the  fraction, 

1      _1_3 

will  be  less  than  the  true  value  of  the  continued  fraction. 

Similar  reasoning  will,  evidently,  hold  in  respect  to  any 
number  of  terms ;  and  will  apply  equally  to  the  general 
form  (2),  as  to  the  particular  example  we  have  considered. 

Hence, 

If  we  include  in  the  reduction  an  odd  number  of  partial 
fractions,  the  result  will  be  too  great  ;  if  an  even  number, 
the  result  will  be  too  small. 

11  1  . 

a.)    The  fractions, — ,        -,         -.  &c, 

2  a    -4-  — - 

are  approximate  values  of  the  given  fraction  ;  and  are 
sometimes  called  approximating  or  converging  fractions, 
or  simply,  convergents. 

b.)  It  is  evident,  that  the  true  value  of  the  continued 
fraction,  lying  between  two  successive  approximate  values, 
differs  from  either  of  them  less  than  they  differ  from  each 
other. 

§  408.   We  have  —  =  — ,  1st  approx.  value. 


ax       ax 


1  '     fl^aa  +  l' 

«i  +  — 


2d    « 


«2 


a*  +  aZ 


a2  +  — 


0203  +  1  3d 


(.ala9  +  1)a3+al 


3'2&  CONTINUED  FRACTIONS.  [§  4Q8. 

We  shall,  evidently,  find  the  fourth  approximate  value, 
or  convergent,  by  substituting,  in  the  third,  a3-\ for  a3. 


«4 


Thus, 

flu«_  -4-  1  \a  .  A-  ci- 
VS,  the  fourth  conver- 


(a2a3-|-l)a4-[-«2 


gent. 

We  find,  obviously,  the  numerator  and  denominator  of 
the  third  convergent,  by  multiplying  those  of  the  second  by 
the  third  partial  denominator,  and  adding  those  of  the  first 
convergent. 

We  find,  in  like  manner,  the  fourth  convergent  from  the 
terms  of  the  second  and  third. 

To  show  the  generality  of  this  law,  let  it  be  admitted  to 
hold  good  as  far  as  the  rath  convergent  (i.  e.  the  conver- 
gent corresponding  to  an). 

'  L    M     N        ,  P   .     ^ 
Let  also  jy,  —-,  -^,  and  —  be  the   convergents  cor- 
responding to  ot„_2,  a„-u  #«i  and  an+x. 

Then,  since  the  nth  convergent  is  formed  according  to 

the  above  law,  we  shall  have  -=-.  =  ,/r. "    ,   -j-..  (3\ 

N'       M'an-\-L'  v  / 

N  1 

If  now  we  substitute  in  — ,  an-\ for  an,  we  shall, 

JS'  «;i-f  x 

P 

obviously,  find    -^.     Thus, 

p_  _  M(""+~^r)+L    (jfc„+z)g„+1+ii/ 

P         Nan+,  +  M 
or,  from  (3),       y,  =  ^— j-y  (4) 

Consequently,  if  the  law  holds  good  for  n  convergents,  it 
will  for  n  -f-  1. 

Hence,  to  find  the  numerator  and  denominator  of  any 


§  409.]  CONVERGENTS.  329 

convergent  after  the  second,  as  the  (n  -f-  l)th,  we  have  the 
following 

RULE. 

§  409.  Multiply  the  numerator  and  denominator  of  the 
nth  convergent  by  the  (n-\-\)th  partial  denominator,  and 
add  to  the  products,  respectively,  the  numerator  and  denom- 
inator of  the  (n  —  l)th  convergent. 

a.)  The  numerator  and  denominator  of  any  convergent 
must  be  respectively  greater  than  those  of  the  preceding ; 
each  numerator  and  each  denominator  being  at  least  equal 
to  the  sum  of  the  two  next  preceding. 

b.)  Moreover,  each  convergent  is  found  by  substituting, 
in  the  preceding,  for  the  last  partial  denominator,  an  ex- 
pression known  to  approach  more  nearly  to  the  true  de- 
nominator. 

Hence,  evidently,  each  convergent  approximates  more 
closely  than  the  preceding  to  the  true  value  of  the  con- 
tinued fraction.     See  §  410.  a,  b. 

1.  Find  the  successive  convergents  of  the  continued 
fraction,  1 


2  + 


1 


l+Wr 

Ans.  \,  £,  §,  t*t,  and  §|£. 

c.)  The  first  four  convergents  are  approximate  values  of 
the  continued  fraction ;  the  last,  %%^,  is  the  true  value. 

d.)  A  continued  fraction  is  sometimes  mixed  (§112)? 
i.  e.  made  up  of  a  whole  number  and  a  fraction.     Thus, 

3  +  1—  («) 

2  + r 


5  +  &c. 

*28 


330  CONTINUED  FRACTIONS.  [§  410. 

In  such  cases,  the  integral  part  may  be  reserved  and  ad- 
ded to  the  convergents ;  or  it  may  be  taken,  with  1  as  a 
denominator,  for  the  first  convergent. 

Thus,  in  the  above  example,  Ave  shall  have  the  conver- 
gents, 8fe  3|,  3if ;  or  f,  |,  V,  lg$. 

1  (6) 


e.)   This  form,  a  -f- 


«i  + 


a2-j-«fec, 

is  sometimes  assumed  as  the  general  form  of  a  continued 
fraction  ;  the  place  of  the  integral  part,  when  it  is  wanting, 
being  filled  with  0. 

In  that  case,  the  jz>st  convergent  is,  evidently,  too  small ; 
the  second,  too  great ;  and  so  on,  those  of  an  odd  order  be- 
ing too  small,  and  those  of  an  even  order,  too  great.  See 
§407. 

Note.    If  the  integral  part  be  zero,  the  first  convergent  will  of 
course  be  o 

§  410.  If  the  second  convergent  of  §408  be  subtracted 
from  the  first,  the  remainder  is  unity  divided  by  the  product 
of  the  denominators.  If  the  third  be  subtracted  from  the 
second,  the  remainder  is  minus  unity  divided  by  the  product 
of  the  denominators. 

Suppose  it  has  been  proved,  that  this  law  extends  to 
n  —  1  convergents  ;  i.  e.  that 

L.  -  —       LM'—L'M  _   ±1 

Z7     M'~z     17M<~     ~T7W' 

M    _N        M      Man  +  L 
M'      N'  ~  M>      M'an  +  L ' 

_L'M—LM'_      LM'—L'M 

the  numerator  of  which  is  the  same  as  that  of  (7),  with  a 
contrary  sign.     Hence,  the   principle  proved  in  regard  to 
the  first  three  convergents,  applies  equally  to  the  whole 
series.     That  is, 
If  each  convergent  be  subtracted  from  that  ivhich  next 


§  411.]       CONVERGENT. — ERROR. — LOWEST  TERMS.      331 

precedes,  the  numerator  of  the  difference  will  be  ±  1 ;  and 
the  denominator  will  be  the  product  of  the  denominators  of 
the  two  convergents. 

a.)  Again,  the  true  value  of  the  continued  fraction  lie? 
between  any  two  successive  convergents,  and  differs  from 
either  of  them  less  than  they  differ  from  each  other  (§  407}. 

M 

That  is,  the  convergent  —  f,  differs  from  the  true  value 

of  the  continued  fraction  by  less  than 


M'N< 
But  (§  409.  a)  M>  <  N> ;  and  .-.   M'~  <  M'N'. 

WW'  <  M™'  That  is' 

Cor.  i.  The  error,  in  taking  any  convergent  whatever  for 
the  true  value  of  the  continued  fraction,  is  numerically  less 
than  unity  divided  by  the  square  of  the  denominator  of  that 
convergent. 

b.)  The  denominator  of  each  convergent  is  greater  than 
the  next  preceding  by  some  whole  number  (§  409.  a). 

Hence,  if  the  fraction  be  infinite,  we  may  find  a  conver- 
gent whose  denominator  shall  be  greater  than  any  given 
quantity ;    and,  consequently, 

Cor.  ir.  "We  may  find  a  convergent,  which  shall  differ 
from  the  true  value  of  the  continued  fraction  by  less  than 
any  given  quantity. 

c.)   Suppose  that  M  and  M'  have  a  common  divisor,  D. 

Then  D  will,  of  course,  divide  L' M  and  LM',  multiples 
of  M and  M1 ;  and,  consequently  (§  102.  Note  c),  the  differ- 
ence of  those  multiples,  LM' — L'M=  ±  1. 

Therefore  D  must  divide  ±  1,  which  has  no  integral  di- 
visor but  unity. 

D  =  \.  That  is, 

Cor.  in.  Every  convergent  is  in  its  lowest  terms. 

§  411.  One  of  the  most  obvious  uses  of  continued  frac- 
tions is,  to  express  approximately,  in  small  numbers,  frac- 
tions whose  terms  are  large.     Thus, 


332  CONTINUED  FRACTIONS.  [§  412. 

i.  £= 


17 

1 

1 

1 

1 

59~ 

w 

T17 

3+17 

Ct) 

3 

•+i 

Here  we  first  divide  both  numerator  and  denominator 
(§113.  3)  of  \l  by  17.  We  then  reduce  ff  to  a  mixed 
number  3TST.;  and,  again,  divide  both  terms  of  T8T  by  8,  and 
reduce  to  a  mixed  number;   and  so  on. 

Evidently,  these  operations  produce  no  change  in  the 
value  of  the  given  fraction. 

a.)  Now  the  several  convergents  of  the  continued  frac- 
found,  are       ^,  f ,  and  J|. 

1  191 

AVe  find  -  =  -~,  too  great ; 

2  16fi 

-  =  — -,  too  small,  but  differing  from 
/       oy 

the  true  value  by  only  ;{3. 

2.  If  the  fraction  proposed  had  been  \ f ,  we  should  have 
found 

59_q    I     8    -q_L    l    --J.1 

__3+-_3+-_o+— T; 

¥  2  +  8 

and  the  convergents,     3,  §,  and  -Jf .  §  409.  d. 

b.)  This  reduction  of  a  common,  to  a  continued  fraction, 
is,  evidently,  effected  by  applying  to  the  terms  of  the  given 
fraction  the  process  of  finding  the  greatest  common  divisor  ; 
the  several  quotients  forming  the  successive  partial  denom- 
inators. 

§  412.  If  it  be  required  to  transform  any  quantity  what- 
ever, x,  into  a  continued  fraction,  the  nature  of  continued 
fractions  will  sufficiently  indicate  the  following 


RULE. 

1.   Find  the  greatest  integer  contained  in  x,  and  denote 


§  413.]       REDUCING  TO  A  CONTINUED   FRACTION. 


ooo 


it  by  a ;  and  denote  the  fractional  excess  of  x  above  a  by 

1  1  1       ^  , 

— .     Then    x  —  a-\ .     .•.:*:.,  = >  1. 

x1  xx  x —  a 

2.  Find  the  greatest  integer  contained  in  xlt  and  denote 
it  by  ax  :  -and  denote  the  fractional  excess  of  xx  above  ax 

by  — .     Then  xx  =  ax  -{-■ — . 

3.  Apply  the  same  process  to  x2,  and  so  on. 

Thus, 

x=za  +  —  =  a-\ =a-\ — - 

«x+  —  ax-] 1 

2  a2  -) .  &.c. 

a.)  If  x  <  1,  we  shall  have  a  =  0. 

h.)  We  shall  always  have  xx,  Xc,,  &c.  >  1. 

For  if  xx  = ,  or  <  1,  we  have  —  =  or  >  1 ;   and  a  is 

xx 

not  the  greatest  integer  contained  in  x. 

c.)  Whenever  we  find  a  denominator,  xn,  equal  to  a 
whole  number,  we  shall  have  xn=.  an  ;  and  the  continued 
fraction  will  terminate. 

This  will  happen,  if  the  cpiantity,  x,  can  be  exactly  ex- 
pressed by  a  common  fraction. 

d.)  If  the  quantity  is  not  equal  to  a  common  fraction 
(i.  e.  if  it  is  incommensurable),  the  continued  fraction  Avill 
extend  to  infinity. 

§  413.  1.  Given  n  =  3.14159  (§  247.  N.  q),  employing 
only  five  decimal  places.  Reduce  n  to  a  continued  frac- 
tion, and  find  approximate  values. 

„   ,    1 
Ans.  it=3-\ 


15  +  * 


Convergents  (§  400.  e),   3,    £,    fff,   Hlb  &c- 


334:  CONTINUED  FRACTIONS.  [§  414. 

Note.  The  second  approximate  value,  22  was  found  by  Ar- 
chimedes; the  fouith,  54^,  by  Adrian  Metius. 

2.  The  common,  or  tropical  year  consists  of  365.242  241 
mean  solar  days.     Find  approximate  values  for  this  time. 

Ans.  365i,  365^,  365383,  365TW,  &c. 
Note.  The  third  approximation  shows  an  excess  of  the  solar 
year  above  365  days,  of  JL  of  a  day.  To  preserve  the  coincidence 
between  the  solar  and  civil  year,  therefore,  eight  years  in  thirty-three 
must  contain  366  days  each.  That  is,  a  day  must  be  added  to  every 
fourth  year  seven  times  in  succession,  and,  the  eighth  time,  to  the 
fifth  year. 

3.  The  sidereal  month  (i.  e.  the  time  of  the  moon's  side- 
real revolution)  consists  of  27.321  661  days;  or,  the  moon 
revolves  1  000  000  times  in  27  321  661  days.  Find  approx- 
imate  values  of  this  ratio.      Ans.  27,  8/,  7^,  3T\°3',  &c. 

Note.  These  ratios  show  that  the  moon  revolves  about  3  times 
in  82  days;  28  times  in  765  days;  or,  more  exactly,  143  times  in 
3907  days. 

§  414.  Continued  fractions  are  also  employed  in  finding 
the  roots  of  equations,  and  in  extracting  the  roots  of  num- 
bers. 

1.  Extract  the  square  root  of  3  ;  i.  e.  find  a  root  of  the 
equation,  x2  —  3  =  0.  (1) 

Here  x  ■=  1  -I . 

xx 

Diminishing  the  roots  of  (1)  by  1  (§  367),  we  have 

y*  +  2y-2  =  0,  (2) 

an  equation,  whose  roots  are  equal  to  — . 

Transforming  (2)  by  §  370,  we  find 

2x?  —  2xr  —  1  =  0.  (3) 

This  gives  xY  =  1  -| . 

•Ms  n 

Transforming  (3)  in  the  same  manner  as  (1),  we  have 

1 
x„2—  2x2—  2  =  0;  (4)      and  x2  =  2  -\ . 

X3 


§  414.]  ROOTS  OF  EQUATIONS.  33") 

We  find,  in  like  manner, 

2aJ32—  2xz—  1  =0,  (oi 

which  being  the  same  as  (3),  will  have  the  same  roots,  and 
will  give  rise  to  transformed  equations  like  (4)  and  (5). 

Hence,  we  shall  have  a  repetition  of  the  equations  (3) 
and  (4),  and  of  their  roots  of  which  1  and  2  are  the  inte- 
gral parts,  in  endless  succession. 

x  —  1  +- -  =  1.732  &c. 

1  + 

2  +  - 


l  +  i&c. 

The  convergents  are  &   f,   f,   f,   \\,   ff,   ||,    ||. 

a.)  A  continued  fraction  of  this  kind,  in  which  any  num- 
ber of  the  partial  denominators  are  continually  repeated  in 
the  same  order,  is  called  periodic. 

b.)  It  will  be  found,  that  every  incommensurable  root  of 
an  equation  of  the  second  degree  may  be  expressed  by  a 
periodic  continued  fraction. 

Of  course,  when  the  first  period  is  found,  such  a  fraction 
may  be  developed  to  any  extent,  by  simply  repeating  the 
period. 

2.    Extract  the  square  root  of  2. 

Convergents,   \,    §,  £,   \l,    §$,   f§,  &c. 


ERRATA.  \ 

Page  14,  first  line,  for  "  16  "  read  "  11." 
"    60,  line  27,  for  "  +  b',"  read  "  —  ft/." 
"    62,  line  19,  for  "  58,"  read  "  28." 
"    69,  last  line,  for  "  +  b  "  read  "  +ab." 
"    83,  first  line,  for  "  +  63"  read  "_&3." 
"    91,  line  15,  for  "  a  =6"  read  "6  =  a;"  and 

for  "a" — 6""  read  "a" — a'1." 
"    92,     "  20,  for  «+ab  "  read  "—  ab." 
"    93,     "  27,  after  "  them  "  insert  "  taken  as  a  divisor.,'> 
"    96,     "  26,  for  "5aZ>3"  read  "  5«36." 
"    99,     "  30,  and  page  100,  line  26,  for  "  12  "  read  "  11." 
"  106,     "   17,  for  "  dividing  "  read  "  multiplying." 
"  119,     "     5,  for  "  13_y  read  "  23^." 

"  128,  "  25,  for  "(2.32)"  read  "(2.32):?." 

"  148,  "  11,  for  "  a—b  "  read  "  a^—b." 

"163,  "     4,  for  "^(^2^2)"  read  "y(p2_?2)-» 

"192,  "     3,  for  "10"  read  "11." 

"198,  "     3,  for  "(1)"  read  "(3)." 

"  292,  "  14,  after  "  have,"  insert  "(x  being  >  1)." 

"  310,     "  11,  for  "+"  read  "-." 


r  it 

It, 


- 


jiliiliili 


